B S Sathyaprakash
email: b.sathyaprakash@astro.cf.ac.uk

B F Schutz
email: b.f.schutz@aei.mpg.de

C Van

Den Broeck
email: vdbroeck@nikhef.nl

Cosmography with the Einstein Telescope

Keywords: numbers: 04.30.Db, 04.25.Nx, 04.80.Nn, 95.55.Ym

Introduction

Standard candles are used to measure the geometry and dynamics of the Universe. A standard candle is a source whose intrinsic luminosity L can be inferred from the observed properties such as its spectral content, time-variability of the flux of the radiation it emits, etc. Since the observations also measure the apparent luminosity F, one can deduce the luminosity distance D L to a standard candle from 4πD 2 L = L/F. If the red-shift z to the source is known, then from a population of such sources it will be possible to measure the cosmological parameters since the luminosity distance is related to the red-shift, in a flat Universe, via

D L (z) = c(1 + z) H 0 z 0 (1 + z ) -3/2 dz [Ω M + Ω Λ (1 + z ) 3w ] 1/2 , (1)
where H 0 is the Hubble constant, Ω M and Ω Λ are the dark matter and dark energy densities at the present time, and w is the dark energy equation-of-state parameter.

There is no unique standard candle in astronomy that works for all distances. A distance scale is built by using several candles, each of which works over a limited range of the distance. For instance, the method of parallax can determine distances to a few kpc, Cepheid variables up to ten Mpc, the Tully-Fisher relation up to tens of Mpc, the D n -σ relation up to hundreds of Mpc and Type Ia supernovae up to redshifts of a few [1]. This way of building a distance scale has been referred to as the cosmic distance ladder. For cosmography, a proper calibration of distances to high red-shift galaxies is based on the mutual agreement between the different rungs of this ladder. It is critical that each of the rungs is calibrated with as little an error as possible.

Self-calibrating standard sirens of gravity

Cosmologists have long sought standard candles that can work over large distance scales without being dependent on the lower rungs of the cosmic distance ladder. In 1986, it was discovered [2] that gravitational astronomy can provide such a candle, or, more appropriately, a standard siren, in the form of an inspiraling compact binary consisting of neutron stars and black holes. This method needs no calibration at all, relying purely on the modelling of the two-body problem in general relativity. Gravitational-wave observations, therefore, provide not only a powerful distance measuring tool but also a useful check on other distance ladders.

In simple terms, in the inspiral phase of a compact binary, the amplitude of gravitational waves depends on the ratio of a certain combination of the component masses called the chirp mass and the luminosity distance. Both of these can be measured for a chirping signal -a signal whose frequency increases by a measurable amount during its observation. For such sources one can directly infer the luminosity distance. In reality, the response of an interferometer to such a signal, in an approximation that keeps only the dominant signal harmonic at twice the orbital frequency, is given by

h(t) = 4A(θ, ϕ, ψ, ι)νM [πM F (t)] 2 3 cos Φ(t) D L . (2)
Here 0 ≤ A ≤ 1 is a numerical factor that depends on the location of the binary on the sky and its orientation relative to the detector, M and ν are, respectively, the binary's total mass and symmetric mass ratio, Φ(t) and F (t) are the signal's phase and frequency, D L is the luminosity distance, (θ, ϕ) give the source's location on the sky, ι is the orientation of the system relative to the line-of-sight, and ψ is the wave's polarization angle. In the case of non-spinning binaries on quasi-circular orbits, an inspiral signal is characterized by nine parameters in all: (M, ν, t 0 , Φ 0 , θ, ϕ, ψ, ι, D L ).

Here, t 0 and Φ 0 are fiducial parameters defining the epoch when the signal's frequency reaches a certain value and its phase at that epoch. In this approximation, the signal amplitude depends only on the chirp mass M defined by M = ν 3/5 M and not separately on the two mass parameters. The signal's phase has been computed to high order in post-Newtonian theory [3] and depends only on the two mass parameters; here we go to 3.5PN. One can therefore employ matched filtering to extract the signal and measure the two mass parameters (M, ν) as well the two fiducial parameters (t 0 , Φ 0 ). Note that for a source at a redshift z the signal's frequency will be red-shifted to F → F/(1 + z) but the mappings M → (1 + z)M and D L → (1 + z)D L leave the signal invariant. Thus, a source of intrinsic total mass M i will appear to be a binary of total mass M = (1 + z)M i . One must optically identify the host galaxy and measure its red-shift to deduce its intrinsic mass M i .

Multi-messenger cosmology with binary neutron stars and GRBs

In general, the response of a single interferometer will not be sufficient to disentangle the luminosity distance from the angular parameters. An optical identification of the source can determine the direction to the source (and its red-shift), leaving three unknown parameters (ψ, ι, D L ). Additionally, if the signal is associated with a gamma-ray burst (GRB), then the source's orbital plane will be perpendicular to the line-of-sight, implying that ι 0 and ψ 0. Thus, for inspirals detected in coincidence with GRBs, a single detector is good enough to measure the luminosity distance. Multi-messengers like GRBs can, therefore, make precision cosmography possible, without the need to build a cosmic distance ladder.

For signals not associated with a GRB, D L can be measured if the source lasts long enough to cause a modulation in the signal's frequency due to the motion of the detector relative to the source, as would be the case for the Laser Interferometer Space Antenna (LISA). If the signal lasts for only a short time, as would be the case for ground-based detectors, one would need a network of three detectors to measure D L .

Over the next two decades GW detectors could provide a new tool for cosmology. The Laser Inteferometer Gravitational-Wave Observatory (LIGO) in the USA and Virgo in Europe have reached design goals for their initial operation. Recent science runs of LIGO and Virgo have begun to impact our understanding of astrophysical sources and phenomena. They are both now getting ready to upgrade to advanced sensitivities by 2014 and are expected to detect ∼ 40 binary neutron star (BNS) mergers each year [4,5]. Red-shift could be measured to a (small) number of events associated with GRBs and might allow the measurement of the expansion rate of the Universe in the ∼ 500 Mpc range, where optical data is scarce [6,7]. Observation by LISA of extreme mass ratio inspirals could measure the Hubble constant pretty accurately [8]. LISA will also observe binary super-massive black hole mergers with signal-to-noise ratios (SNRs) of several thousands at z ∼ 1, enabling the measurement of the dark energy equation-of-state to within several percents [9,10].

In the rest of this paper we will discuss how well it might be possible to constrain cosmological parameters with the Einstein Telescope (ET) -a third generation ground-based interferometer that is currently under a design study [11,12]. ET is envisaged to be ten times more sensitive in amplitude than the advanced groundbased detectors, covering a frequency range of 1-10 4 Hz. Achieving the sensitivity of ET will pose challenges in mitigating gravity gradient, thermal and quantum noises but careful analysis shows that the technology might be within reach in the next decade [13]. One possible topology for ET could be an equilateral triangle, ten km on a side, allowing the operation of three V-shaped interferometers at a single site [14,15]. A network of such detectors might be available over the next 15-20 years, but we shall explore how accurately it might be possible to measure the cosmological parameters with a single such ET.

Distance reach of ET for neutron star binaries and event rates

For a fixed SNR ρ 0 , the distance up to which an inspiral signal could be detected in ET is given by

D L (ρ 0 ) = A (θ, ϕ, ι, ψ)M 5/6 ρ 0 F lso Fs f -7/3 S h (f ) df 1/2 , (3)
where S h (f ) is the one-sided noise power spectral density (PSD), which we take to be the "ET-B" PSD as in Fig. 1 of [15]; we also assume the 10 km triple Michelson set-up as explained in that reference. F s is a frequency below which the accumulated SNR is negligible because the PSD rises far faster than the signal spectrum and F lso is the signal frequency corresponding to the last stable orbit of the binary, taken to be F lso = 1/(6 3/2 πM ). The quantity A is an appropriate combination of the function A in Eq. ( 2) and the antenna pattern functions of the 3 independent Michelson interferometers. In computing the distance reach of ET one can take an average of A over sky position and set ι = ψ = 0, again assuming relatively strong beaming of GRBs; the corresponding root-mean-square (rms) value is A rms = 2/5A opt , where A opt is the value for an optimally oriented and positioned system [16]. However, we note that GRBs may have beaming angles up to ∼ 40 • , i.e., ι ∼ 20 • [17]. If one angle-averages over all angles (θ, φ, ψ, ι) but with the constraint ι < 20 • , then the rms value becomes A rms = αA opt with α 0.614, which is barely different from the case ι = ψ = 0. The distance up to which ET might detect signals from inspiraling BNS with an SNR of 8 is shown in Fig. 1 as a function of the observed total mass (blue dashed lines). We assume a cosmological model in which H 0 = 70 km s -1 Mpc -1 , Ω M = 0.27, Ω Λ = 0.73 and w = -1, which allows us to convert distances to red-shifts by inverting Eq. ( 1), and which we have used to convert from the observed masses in Fig. 1 to the intrinsic masses (red solid lines). By extrapolating the rate of BNS inspirals expected in advanced detectors to ET, whose distance reach for BNS is z 2 or D L 16 Gpc, one might expect 4 × 10 7 events per year. Of course, this naive extrapolation does not give the correct rate as it does not include the cosmological evolution of compact binaries. For our purposes, however, even if the rate is an order of magnitude lower it does not matter. As an aside, legitimate concern has been raised that this high event rate may lead to a confusion background [18]. As it turns out, the PSD-weighted signals are actually quite short and won't tend to have significant overlap with each other [19]. As noted earlier, in order for BNS inspirals to be useful for cosmography, it is essential that their location on the sky and red-shift are determined separately. If, as suspected, BNS are progenitors of short-hard GRBs [17], then it might be possible to make a coincident detection of a significant subset of the events in GW and electromagnetic (EM) windows.

Since GRBs are believed to be beamed with beaming angles of order 40 • , we assume that only a small fraction (∼ 10 -3 ) of binary coalescences will have GRB or other EM afterglows that will help us to locate the source on the sky and measure its red-shift. Eventually, we will be limited by the number of short-hard GRBs observed by detectors that might be operating at the time. As a conservative estimate, we assume that about 1, 000 BNS mergers will have EM counterparts over a fiveyear period. For definiteness we consider only BNS mergers and take these to have component masses of (1.4, 1.4)M .

Measurement accuracy of cosmological parameters

How well would we measure cosmological parameters with a catalogue of such sources? To answer this question we simulated 5,190 realizations of the catalogue containing 1,000 BNS coalescences with known red-shift and sky location, but with the luminosity distance subject to statistical errors from GW observation and weak lensing. We assumed that the sources were all in the red-shift range 0 ≤ z ≤ 2, distributed uniformly (i.e., with constant comoving number density) throughout this red-shift range. The luminosity distance to the source was computed by assuming an FRW cosmological model with H 0 = 70 km s -1 Mpc -1 , Ω M = 0.27, Ω Λ = 0.73, and w = -1, but the measured distance was drawn from a Gaussian distribution whose width σ DL was determined by the quadrature sum of the errors due to weak lensing and GW observation. Weak lensing error in D L was assumed to be 5% at z = 1 [20] and linearly extrapolated to other red-shifts. The GW observational error was estimated from the covariance matrix C km of the parameter space of the unknown signal parameters p k :

C km = Λ -1 km , Λ km = h k , h m , h k = ∂h ∂p k . (4)
Here the angular brackets denote the scalar product, which, for any two functions a(t) and b(t), is defined as

a, b = 4 ∞ 0 df S h (f ) A(f ) B * (f ) (5)
where A and B are the Fourier transforms of the functions a(t) and b(t), respectively, and S h (f ) is the ET noise power spectral density. In computing the covariance matrix, we made a distinction between two cases:

• If the beaming is so strong that one can assume ι = ψ = 0 for all practical purposes, then the parameters to be estimated are

p k = (M, ν, t 0 , Φ 0 , D L ). ( 6
)
At a given redshift, the resulting rms uncertainty on D L was then averaged over sky position. • Allowing for beaming angles as wide as ι ∼ 20 • , the binary's orientation (ψ, ι) also needs to be taken into account:

p k = (M, ν, t 0 , Φ 0 , D L , ψ, ι). (7)
In this case the rms errors were averaged over all angles (θ, φ, ψ, ι) but with the constraint ι < 20 • .

We then fitted each realization of the source catalogue to the cosmological model given in Eq. ( 1), using the Levenberg-Marquardt algorithm [21,22], in order to find a set of best fit parameters. It turns out that a catalogue of 1,000 sources is not quite enough for an accurate determination of all the parameters. However, assuming that H 0 is known accurately, the algorithm gave the best fit parameters in (Ω M , Ω Λ , w) for each of the 5,190 realizations. By the time Einstein Telescope is operational, a network of advanced detectors will have been active over a number of years, possibly measuring H 0 down to a percent or better over such a time span [7]. Using extreme mass ratio inspirals seen in LISA, one can also measure H 0 to better than a percent [8], or down to a fraction of a percent judging from relative distance errors for supermassive binary black hole coalescences [10]. Taking H 0 to be the only free parameter and with only 50 sources up to a redshift of z = 0.5, ET itself would achieve an accuracy of 0.55% (using the kind of estimation explained here). For sufficiently low-redshift sources, the rest of cosmology is not very important. Having determined H 0 in this way, the higher-redshift sources can then be used to explore the cosmological parameter space more fully. In this way, GW astronomy will provide an independent measure of cosmography.

Thus, ET itself can use sources in the low-redshift Universe to measure H 0 with an error that is negligible compared to the uncertainties on the other parameters obtained from all sources combined, as we shall see below. For the purposes of this paper we will consider H 0 essentially known.

Let us first discuss some results in the case of strong beaming (ι = ψ = 0); in the next section we will relax this assumption. The distributions P of the parameters obtained in the way we described above are shown in Fig. 2, where the vertical line

σ Ω Λ = 0.031 σ Ω M = 0.045 <Ω M > = 0.260 σ Ω M = 0.035 <Ω Λ > = 0.736 σ Ω Λ = 0.026 <w> = -0.96 σ w = 0.15 Ω M Ω Λ w Figure 2.
The plot shows the distribution of errors in Ω M , Ω Λ and w, obtained by fitting 5,190 realizations of a catalogue of BNS merger events to a cosmological model of the type given in Eq. ( 1), with three free parameters. The fractional 1-σ width of the distributions σ Ω M /Ω M , σ Ω Λ /Ω Λ , and σw/|w|, are 18%, 4.2% and 18% (with weak lensing errors in D L , left panels) and 14%, 3.5% and 15% (if weak lensing errors can be corrected, right panels).

is at the true value of the relevant parameter. Figure 3 shows a scatter plot of the recovered values of the dark energy density and dark energy equation-of-state when not correcting for weak lensing errors.

The relative 1-σ errors in Ω Λ , Ω M and w, are 4.2%, 18% and 18% (with weak lensing) and 3.5%, 14% and 15% (with weak lensing errors corrected). Although P(w) is quite symmetric, P(Ω M ) and P(Ω Λ ) are both skewed and their mean values are slightly off the true values. The medians, however, are coincident with the true values.

In addition to H 0 if Ω Λ is also known (or, equivalently, if Ω M + Ω Λ = 1), then one can estimate the pair (Ω M , w) more accurately, with 1-σ errors in Ω M and w, of 9.4% and 7.6% (with weak lensing) and 8.1% and 6.6% (with lensing errors corrected). Finally, if w is the only parameter unknown, it can be measured to an even greater accuracy with 1-σ errors of 1.4% (with weak lensing) and 1.1% (with lensing errors corrected). ‡ ‡ At this point we note that in contemporary cosmology, w is determined mainly through SNIa observations using CMB data as a prior to "fix" the other parameters. The CMB constraint on w is extremely weak. If one were to use CMB results as a prior for GW measurements, one would obtain an independent measurement of w. We stress once again that, unlike supernovae, GW standard sirens do not need any external calibration. A detailed discussion will be presented in forthcoming work [START_REF] Wen | Work in progress[END_REF].

Effect of unknown orientation and polarization

In the previous section our study neglected the effect of different inclinations of the orbit to the line-of-sight. Varying the inclination has two distinct effects: On the one hand, as noted in Ref. [7], due to the strong correlation between the luminosity distance and inclination, the estimation of luminosity distance could get corrupted. On the other hand, binaries that are not face-on are, in general, elliptically polarized and have a non-zero polarization angle. Since polarization angle is correlated with the luminosity distance, there could be further degradation in the estimation of the luminosity distance.

In this section we will relax the condition that the inclination of the orbit is precisely known. However, we shall restrict the inclination of the binary's angular momentum with the line-of-sight to be within 20 degrees. We shall also assume that the radiation is described by an arbitrary polarization angle. Since the sky position is still assumed known, this gives us a 7 × 7 covariance matrix with a revised estimate for the error in the luminosity distance. As before, we construct catalogs of binary coalescence events but with the luminosity distance now drawn from a Gaussian distribution with revised widths. We fit each catalog to a cosmological model and then repeat the exercise 5,190 times to estimate the accuracy with which the various cosmological parameters can be measured.

As expected, the parameter measurments get worse if we assume two or more parameters to be unknown. For instance, errors in the estimation of Ω M , Ω Λ and w, are, respectively, > 100%, 24% and 47% with weak lensing and > 100%, 21% and 43%, if weak lensing can be corrected. Similarly, if Ω Λ is assumed to be known then the errors in the estimation of Ω M and w are, respectively, 12% and 9.5% if weak lensing is uncorrected for and 11% and 9.2% if weak lensing can be corrected. However, the results are more or less the same if dark energy parameter w is the only unknown quantity. Even when the inclination and polarization angles are taken as free parameters, but inclination angle is restricted to within 20 degrees, the error in the estimation of w is 1.4% with weak lensing and 1.3% if weak lensing can be corrected.

Comparison with other experiments and space missions

These results are better than what could be achieved with LISA, which in estimating w as a single free parameter may reach an accuracy of 0.3-10% depending on the systems available, but on condition that weak lensing effects can at least partially be subtracted [10], otherwise LISA's error will be much larger (in the order of 20%). Our results where multiple parameters are estimated at once are also competitive with what is deemed possible with proposed dedicated dark energy missions; see, e.g., the report of the Dark Energy Task Force [START_REF][END_REF], who expect to see w errors between 3.5% and 11%.

In the above we computed uncertainties for at most three free parameters. When (H 0 , Ω M , Ω Λ , w) are all estimated at the same time, the relative 1-σ errors come out to be 13%, 37%, 36%, and 23%, respectively. We stress once again that ET can use a relatively small number of sources at low redshifts to obtain H 0 with almost negligible uncertainty; this parameter can then be considered fixed to study (Ω M , Ω Λ , w) over a much larger redshift range where the dynamical history of the Universe truly comes to bear.

Conclusions

In this study, we have shown that using a population of binary neutron stars Einstein Telescope can measure the dark energy equation-of-state parameter w (assuming this is the only unknown parameter) to within about 1.5%. This precision is achievable even if it is not possible to correct for the errors in the luminosity distance caused by weak lensing and the unknown inclination and polarization angles of the binary.

The results of our simulation are quite encouraging but further work is needed to confirm the usefulness of GW standard sirens in precision cosmology. Let us mention some that are currently being pursued. Spins of component stars can be legitimately neglected in the case of neutron stars (and hence in BNS) but not for black holes. The modulation in the signal caused by the spin of the black hole can improve parameter estimation accuracies [START_REF] Vecchio | [END_REF]. We assumed, for simplicity, that all our sources are BNS systems with masses (1.4, 1.4)M . In reality, the catalogue will consist of a range of NS and BH masses. A more realistic Monte Carlo simulation would draw binaries from the expected population rather than the same system, some of which (e.g. more massive systems) would lead to better, but others to worsened, parameter accuracies. The signal contains additional features, such as other harmonics of the orbital frequency than the second harmonic considered in this work, and the merger and ringdown signals. These are important for heavier systems and could potentially reduce the errors. Also, a recent study [26] has shown that weak lensing effects on the measurement of the luminosity distance could be reduced by 50% , which should make for improved parameter accuracies. These factors are currently being taken into account to get a more reliable estimation of the usefulness of ET in precision cosmography.

Figure 1 .

 1 Figure1. Range of the Einstein Telescope for inspiral signals from binaries as a function of the intrinsic (red solid line) and observed (blue dashed line) total mass. We assume that a source is visible if it produces an SNR of at least 8 in ET.

Figure 3 .

 3 Figure 3. Scatter plot of the retrieved values for (Ω Λ , w), with 1-σ, 2-σ, and 3-σ contours, in the case where weak lensing is not corrected.


Jonathan Piat

François Moreews
email: fmoreews@irisa.fr

Olivier Collin

Dominique Lavenier

Alexandre Cornu

SLICEE: A Service oriented middleware for intensive scientific computation A bioinformatics case study

Keywords: Orchestration, SOA, service, HPC, parallelism extraction, middleware I

With the growth of computational power and storage demand from bioinformatics applications, cluster computing has emerged as a good answer. Bioinformatics applications are often structured as workflows that are composed of a set of operations to perform on large data sets. These workflows are deployed as complex script that handles the sequence of program calls with their relevant inputs and try to take advantage of a computer cluster using a scheduler. Their performance rely on the user ability to analyze the potential parallelism in the workflow. SLICEE (Service Layer for Intensive Computation Execution Environment) proposes to abstract the calls to the cluster scheduler by handling command submission and data management. A workflow client is used to orchestrate the SLICEE services, that takes care of exploiting the data parallelism, and data routing between tasks to take advantage of the cluster architecture. A storage management layer takes care of optimizing data transfer with a reference passing mechanism. More-over SLICEE handles multi site command submission and data storage in a way that makes it transparent to the end-user. This tool enables an easy to maintain and to share implementation of bioinformatics workflows using intensive computation resources.

INTRODUCTION

Bioinformatics applications such as tools for NGS (Next Generation Sequencing) data processing challenges today's computation resources by raising the amount of data to process and computation complexity to a new level. The common answer to such challenge is to use a computer grid or computer cluster with a large storage facility. Computer clusters combine the processing power of multiples machines that can be addressed as a single computing resource. A scheduler such as SGE (Sun Grid Engine), PBS (Portable Batch System), TORQUE or Condor is then in charge of dispatching the processing demand onto the available processing resources. Such architecture allows to take advantage of coarse grain parallelism to speed-up computation of scientific applications. This parallelism can either be used by running multiple applications in parallel or by designing the application in such way that it can be divided into smaller grain tasks to be achieved in parallel.

Bioinformatics applications can usually be described as a sequence of operation to perform on an input data-set. Such sequence of operations can be modeled using a workflow/dataflow model. This class of model represents the application as a network of operations connected through their data-dependencies. Such representation allows to retrieve the available parallelism in the application by analyzing data dependencies. Bioinformatics workflow are often deployed as a set of complex hand written scripts that represent the sequence of operations and their submission to the cluster scheduler with manual management of data dependencies between calls. These scripts are hard to maintain and their performance rely on the users ability to take advantage of the available data parallelism within the application.

Graphical workflow engines such as Kepler [1] propose to ease workflow specification through a graph editing UI. This tool also propose to schedule and run the workflow even addressing clusters through scheduler (SGE, Torque, PBS...) or Opal web service. Tasks are then achieved by uploading inputs, performing the task and download the outputs thus doubling the data transfer overhead. This makes it hard to deal with distributed data intensive applications, usually dealing with Gbytes of data. Kepler does not propose automatic data parallelism extraction from the workflow structure, but only rely on the user ability to represent it.

The SLICEE (Service Layer for Intensive Computation Execution Environment) tool proposes to address the parallelism available on a computer cluster with minimum user intervention. Its service oriented approach handles job submission with automated data parallelism extraction. Moreover this service permits multi site job submission and distant data storage on a non-shared file system with no user intervention. When running a workflow through SLICEE, input data are uploaded once if local and then final output can be retrieved without partial result downloading, thus limiting the required bandwith and data transfer overhead. SLICEE also features a user authentication that allow one user to manage its own data sets and execute applications with its access rights thus preserving existing access and quota policy.

This tool can be supported by most workflow engines. In this paper we demonstrate the use of SLICEE using the Kepler workflow framework for graphical workflow edition, and tasks orchestration. In this approach Kepler is used to orchestrate the workflow but tasks execution is delegated to SLICEE with only references on data being exchanged between workflow tasks.

First we start with a describtion of the application context by giving a brief overview of the bioinformatics context and its challenging applications. A second part presents existing workflow execution systems while a third part presents SLICEE followed by an application example. In a final part we conclude and give directions for future work and improvements.

II. BIOINFORMATICS APPLICATIONS

Bioinformatics applies computational approaches to study molecular biological entities like genes and proteins. Many algorithms found in the bioinformatics field are computationally intensive and process a huge amount of data. To illustrate, the recent technological breakthrough in sequencing technology allows to decipher a genome sequence for an affordable price but has drastically increased the requirement of high performance computing resources and data storage. In an academic context, bioinformatics analysis usually consists in a script calling heterogeneous specialized software provided by the research community. Even if more and more tools like mpiBlast [START_REF] Darling | The Design, Implementation, and Evaluation of mpiBLAST[END_REF] use distributed computational resources like cluster-nodes or CPU-cores, the data parallelism pattern should often be manually coded. Many bioinformatics programs are command line based and can easily be wrapped as web services with dedicated toolkit like Opal [START_REF] Krishnan | Opal: SimpleWeb Services Wrappers for Scientific Applications[END_REF]. Although the orchestration of such bioinformatics web services has proved to be realistic in a intensive computation production environment [START_REF] Bailey | MEME Suite: tools for motif discovery and searching[END_REF], the designer still needs to integrate in his workflow many components for data transfer, I/O [START_REF] Li | Building cyberinfrastructure for bioinformatics using service oriented architecture[END_REF] and data parallelism.

Bionformatics intensive computing platform commonly offer access to many users simultaneously. Each of them can run parallel jobs on the cluster-nodes. In this context, a strong authentication mechanism is important to monitor the jobs of each user and preserve user data confidentiality.

III. STATE OF THE ART ON SERVICE ORIENTED WORKFLOW APPLICATIONS FOR BIOINFORMATICS

Building a highly maintainable shared repository of bioinformatics workflows is challenging and can really speed-up genomics research. Multiple tools have emerged to propose service oriented workflow bioinformatics design and execution environment.

Conveyor [START_REF] Linke | Conveyor: a workflow engine for bioinformatic analyses[END_REF], a workflow engine for bioinformatic analyzes, propose a library to design workflow components constructed with a strongly typed, interface driven, object oriented design. These components can wraps existing classes for functionality or integrate existing applications. A custom GUI allows the designer to instantiate these components into a workflow and monitor its execution. The workflow execution is performed through the Conveyor web service that orchestrate the components and execute them on a single machine, but supports inter-components parallelism by making use of multi-threading. This tool does not support access to data on a non shared file-system and currently only support applications execution on a single machine making it improper for use in a intensive computation environment.

The MeDICi Integration Framework (MIF) [START_REF] Gorton | The MeDICi Integration Framework: A Platform for High Performance Data Streaming Applications[END_REF] propose to design components by encapsulating code into MIF components that can then be distributed across a cluster or a network of processing nodes. Components can exchange data either as reference or data copy to minimize data transfers. The components executes in a MIF container that manages a local data cache for inter components communication and allow to connect external MIF resources for remote execution. A Kepler integration was proposed in [START_REF] Chase | Kepler + MeDICi[END_REF] that allow one user to design a Kepler workflow that makes call to MIF components to execute actors processing. This approach allows to exploit fine grain parallelism but impose to write applications using the MIF framework. This makes it unsuitable for bioinformatics legacy applications integration.

Taverna [9] are graphical workflow authoring and execution tools able to orchestrate remote web services or local component. Taverna's strong point is its ability to integrate a large set of Domain specific (BIOMOBY [START_REF] Kawas | BioMoby extensions to the Taverna workflow management and enactment software[END_REF], BIOMART [START_REF] Smedley | BioMart-biological queries made easy[END_REF]), resource specific (OPAL) or generic (WSDL, REST) clients API.

OPAL client API integrated in Taverna and Kepler is designed for bioinformatics intensive computation. It uses WSDL services with an asynchronous call pattern and proposes job scheduler submission capabilities. During the worfklow execution, submitted jobs are anonymized, without personal resource allocation policy. I/O data are passed by values or local file path. In such workflows, the instantiation of many glue component for I/O and data transfer is still necessary. Web based tools like Ergatis [START_REF] Orvis | Ergatis: a web interface and scalable software system for bioinformatics workflows[END_REF], Mobyle [START_REF] Néron | Mobyle: a new full web bioinformatics framework[END_REF] and Galaxy [START_REF] Goecks | Galaxy: a comprehensive approach for supporting accessible, reproducible, and transparent computational research in the life sciences[END_REF] have been successfully adopted to make an intensive production environment available to the end user, the biologist. They are suitable to integrate intensive computation tools and support a large collection of bioinformatics components, and they have orchestration capabilities.

IV. THE SLICEE MIDDLEWARE

The SLICEE aims at providing the user with an easy way to exploit the available parallelism on a cluster to run bioinformatics applications at minimal development cost. A client allow the user to graphically design its application pipeline, and offload tasks execution and data management to the SLICEE service through web-services calls. Executed tasks are command line tools which allows to benefit of a large repository of applications. Data management helps at optimizing data transfer to limit inter-tasks communications overhead. The workflow orchestration is performed by the calling client thus preserving the confidentiality of the workflow structure. This approach makes this tool relevant for both academic and industrial use.

Drawing 1: Overall tool structure. The client orchestrate a workflow that delegates tasks execution to the SLICEE service. This service can either submit the command through a web service (OPAL) or using a direct job submission in a ssh session.

A. SLICEE Features 1) Session Management

A workflow execution always start by opening a session in the application. Users can either be authenticated using an LDAP directory or using a locally stored user account. Once the user is authenticated, he obtains a session token that he can use to call SLICEE services such as job posting, data uploading/downloading. Moreover, when authenticated a user can access the data generated by its previous activities for use in the current session. These sessions can also helps at tracking the user activity for better error recovery and/or application debugging.

A user can also register identities that are domain specific authentication data. These identities can then be used to post job on distant clusters with an ssh authentication or retrieve data from hosts that requires authentication. This allow to execute applications and use data with limited access rights. Identities are not saved on persistent storage for security reason but only stored in ram for the session lifetime.

2) Storage service layer SLICEE has its own data management system for better performance. This data management system allows the use of distant data repository (through scp, http and soon to come S3) with transparent data downloading uploading from the user perspective. Moreover this data management system maximize the data locality while posting job on distant cluster. SLICEE references data through data sets. A data set is a set of files and directory that result from a job execution, a data uploading or a data transformation. A data set is identified by a URI with the "id" scheme, the server name as host and a long unique identifier. This URI can be associated to a query field that can contain a "filter=" query to extract specific data from the data set.

Drawing 2: SLICEE data set URI specification

A data references a file or a directory through a URI. These URI supports several VFS like scheme, that specify data access protocol and host. This allow SLICEE to access datas in a non-shared file system by using several protocol for copy, read and write in a VFS fashion. Thus protocol handling remains hidden to the user.

SLICEE also optimizes data locality to minimize bandwidth used by data transfer by avoiding data copy when possible.

3

) Job posting

The main purpose of the SLICEE is to execute commands using the processing power of a computer cluster. The cluster can either be addressed by a scheduler like SGE, Torque, PBS (only SGE supported at the time) or through a web service wrapping job schedulers, using OPAL. A command submitted to SLICEE consists in a command pattern with variable attributes. These variable inputs are surrounded by "$". Two kind of variable are currently supported:

• A parameter variable is identified by the prefix "PARA" (see Figure [START_REF] Bailey | MEME Suite: tools for motif discovery and searching[END_REF]) and is replace in the command at execution time by the variable value. • A data denotes a variable that references a data set. It is identified by the prefix "DATA". This variable can be associated to a method that specify a processing to be applied on a data set by the server. Two processing are currently supported : splitting and joining (see Figure [START_REF] Bailey | MEME Suite: tools for motif discovery and searching[END_REF]) . The join method, groups the input data as a single input, while the split method cuts the input data into several chunks to be processed in parallel. This split method can take up to three arguments: expr that specify the regular expression to match on cut, size to specify data chunks size, and extension that specify the file extension for data chunks files. Drawing 4: Variables syntax description. The parameter variable is composed of the prefix "PARA" followed by the parameter name and surrounded by $. The data variable is composed of the prefix DATA followed by the data name. An additional optional field can be used to use a processing method on the data set.

The command submission also contains a parameter values list and a data values list containing lists of data sets URIs. Other attributes specify the job posting url (either http for opal, or ssh for SGE) and the queue to post (only supported in SGE) to allows posting on accelerated queues running on FPGA or GPU.

In a first step SLICEE generates a set of command from the command submission. It starts by creating new data sets that contains the result of applying the processing method defined in the command pattern to the data sets pointed by the lists of URI contained in the command submission.

Then it generates a command for each of the data sets pointed by the $DATA..$ variables. At last a command is generated for each of the data in the data set that matches the URI filter (if present).

In a second step generated commands are submitted either through opal or directly to the job scheduler. For the opal submission the opal java API is used to create the submission. For the job scheduler submission a custom client creates job posting scripts that are then uploaded to the submission url with input files (if not already available locally) and launched using a ssh session (opened using user identities). Once a job is posted, the server returns a list of URIs that references the data sets resulting of the jobs execution. This URI can be used to track the jobs status, access jobs results when the job has ended or used as command input for later command submission.

4) Parallelism extraction

One of the main feature of the SLICEE job submission service is that it extracts the available data parallelism to execute one command submission as a set of parallel commands. For single data input commands, parallelizing is straightforward, as as much command as the data set datas can be launched. For a command taking N data inputs and P data sets for each input, N P commands are generated, covering all the possible input combination.

Moreover as the result of a job submission is a list of data sets, following tasks may take advantage of the parallelism revealed by the previous command. For a command that generates N files, the following tasks may run in N parallel command executions.

Drawing 5: Parallel processing of a single command submission with multiple input datasets. A command submission takes a list of datasets as input. A command is created for each element of the list. Moreover a dataset is composed of multiple files that are processed in parallel.

Table 1: Comparison of the workflows execution tools. MIF an SLICEE can be compared due to their common integration with Kepler. Galaxy and Mobyle propose a full web approach while Taverna only focus on discovery and orchestration. Taverna is also an interesting front-end to call SLICEE services. B. Kepler SLICEE client

We implemented a Kepler based client to interface with SLICEE. Kepler is a scientific workflow tool based on Ptolemy. Kepler provides the graphical user interface for workflow edition, and workflow tasks orchestration. Our Kepler plugin implements several actors for SLICEE command submission, data sets filtering and data preview. The command submission actor (called Easy Remote Execution) is a configurable actors that takes the SLICEE formatted command pattern as an attributes and automatically generates actor ports for variables elements. When executed, the actor first upload to the server the input data if local, and then submits the command to the server. When the command ends, the actor only outputs the URIs of the resulting data sets. These data sets can then be used as actors input without downloading their content or previewed (limited in size) within the UI. This client also implements a logging feature to log the workflow execution for testing and recovery purpose. A recovery actor allows to perform error recovery on the client side by using a log file to restore the workflow to a given state. This allow to recover error that may result from a service or connection failure.Multiple example of bioinformatics applications were developed to show the tool capabilities.

Drawing 6: SLICEE propose three services layer to manage the different workflow execution aspects. The authentication layer ensure the user authentication consistency along the workflow execution. The command submission makes use of the available data parallelism to generate the command set to execute. The Data management service, allows the user to upload data, download results and manages the multi-site data reference cache.

V. APPLICATION CASE STUDY

In this example we demonstrate the use of SLICEE through a phylogeny workflow designed using Kepler. This workflow shows a simple phylogenomics application for predicting the function of a protein. Input is a protein sequence which is scanned against the Swissprot protein knowledge base to retrieve sequences with homologies. The resulting set of sequences is thus globally aligned, processed to remove ambiguously aligned regions, then a phylogenetic tree is built.

To accelerate the workflow execution the Blastp and ClustalW operations are accelerated using a FPGA implementation that can be used through a special job scheduler queue. The idfetch, Gblocks [START_REF] Castresana | Selection of conserved blocks from multiple alignments for their use in phylogenetic analysis[END_REF], Phylip [16] tasks runs in general purpose queue. As FPGAs are more efficient when processing large volume of data, input datas of Blastp and ClustalW uses the method "join()" on the data inputs to ensure these tasks runs a single command with a large amount of data. In contrary tasks idfetch, Gblocks and Phylip uses the "split()" method on their inputs with relevant arguments to process multiple data chunks in parallel for better performance.

Illustration 1: Structure of the phylogeny workflow. This example workflow exposes multiple levels of parallelism. Fine grain parallelism is exploited on Blastp and ClustalW using a FPGA implementation. IdFetch Gblocks and Blastp benefits of the coarse grain data parallelism extracted by SLICEE.

This example was tested on the GenOuest cluster that provide computing resources like FPGA, GPU and general purpose high performance cores. All these resources benefits from a shared file-system. This approach permits to exploit both the intrinsic parallelism of an FPGA by feeding it with large volume of data and still take advantage of the coarse grain parallelism available on the cluster for applications executing on general purpose cores. Moreover the data management system of SLICEE using only references on data, minimize the data transfer between tasks. Only two data transfers occurs in this example, one for input sequence uploading and one for final result downloading. In the case of a multi-site execution, transfer would only have occurred on purpose.

VI. CONCLUSION AND FUTURE WORK

The SLICEE middleware is designed to be used on a single cluster but multiple instances deployed on different remote sites can be called by a client within the same workflow execution. It provides a partial lightweight distributed execution environment, far from classical grid computing environment with mechanisms like certificate based authentication and resource sharing but much easier to deploy. Furthermore, the fast increase of requirements for bioinformatics treatment leads to the adoption by bioinformatics computing platforms of an hybrid or private cloud model. Service oriented middlewares like SLICEE will enable a smooth migration of bioinformatics workflows execution from clusters to cloud. Cloud Storage systems have proved to be a valuable option for scientific workflows [START_REF] Juve | Data Sharing Options for Scientific Workflows on Amazon EC2[END_REF]. Thus we plan to integrate the support of cloud storage services like amazon S3 in the storage service layer of SLICEE. The design paradigm of Service-oriented architecture provides important benefits to the bioinformatics field driven by an academic scientific community where shared infrastructures and collaborative development efforts are mainly adopted.

Today bioinformatics web services consists in database content search tools and analysis tools but even if bioinformatics services orchestration editors are attracting interest, their usage remains limited to educational or prototyping purposes but dedicated service layers can bring the power of intensive computation to the end user.

Drawing 3 :

 3 scheme host path query

$

  DATAsequence.split(">+[^>]", 5, ".fasta"

VII. ACKNOWLEDGMENTS

SLICEE is within the BioWIC project (Bioinformatics Workflows for Intensive Computation) , funded by the ANR (french National Research Agency) ANR-08-SEGI-005. We also thank the GenOuest and the Migale bioinformatics platform for hosting the BioWIC tools.


S Raulot

A Savo

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms *

Keywords: AMS 2000: 58J50, 35P15 Manifold with boundary, Differential forms, Eigenvalue, Sharp bounds 1

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.

Introduction

Let Ω be a compact, connected (n + 1)-dimensional Riemannian domain with smooth boundary Σ n . The Dirichlet-to-Neumann operator T , also called Steklov operator, acts on smooth functions on Σ in the following way. If f ∈ C ∞ (Σ) and f denotes the unique harmonic extension of f to Ω, then:

T f = - ∂ f ∂N , (1)
where N is the inner unit normal vector field on Σ. T defines a pseudo-differential operator on C ∞ (Σ) which is known to be elliptic and self-adjoint; hence T has a discrete spectrum 0 = ν 1 < ν 2 ≤ ν 3 ≤ . . . . Note that the lowest eigenvalue is ν 1 = 0, corresponding to the constant eigenfunctions; therefore, in our convention, the first positive eigenvalue of T will be denoted by ν 2 . There is a vast literature on eigenvalue estimates for the operator T ; directly related to our paper are the estimates given in [START_REF] Escobar | The geometry of the first non-zero Steklov eigenvalue[END_REF] and [START_REF] Escobar | An isoperimetric inequality and the first Steklov eigenvalue[END_REF].

In this paper, we consider a natural extension of the Dirichlet-to-Neumann operator T to an elliptic operator T [p] acting on differential forms of arbitrary degree p on the boundary Σ and then prove some geometric lower bounds for its first eigenvalue, given in terms of the second fundamental form of the boundary. We then estimate these new eigenvalues from above in terms of the isoperimetric ratio Vol(Σ)/Vol(Ω), and in terms of the eigenvalues of other differential operators, like the Hodge-Laplace operator on the boundary Σ and the biharmonic Steklov operator. In some cases we improve some known estimates. The operator T [p] seems to have interesting spectral properties which, we hope, justify the present work.

In the rest of the introduction we state the main results of the paper.

1.1 The definition of T [p] Let ω be a form of degree p on Σ n , with p = 0, 1, . . . , n. Then there exists a unique p-form ω on Ω such that: ∆ω = 0 J ⋆ ω = ω, i N ω = 0, where J ⋆ denotes the restriction of ω to Σ, and i N is the interior product of ω with the inner unit normal vector field N. The form ω will be called the harmonic tangential extension of ω. Its existence and uniqueness is proved, for example, in Schwarz [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF]. We set:

T [p] ω = -i N dω,
and then we have a linear operator T [p] : Λ p (Σ) → Λ p (Σ), the (absolute) Dirichlet-to-Neumann operator, which reduces to the classical Dirichlet-to-Neumann operator acting on functions when p = 0, so that T [0] = T . Here Λ p (Σ) denotes the vector bundle of differential p-forms on Σ. We observe in Section 2 that T [p] is an elliptic self-adjoint pseudo-differential operator, with discrete spectrum ν 1,p (Ω) ≤ ν 2,p (Ω) ≤ . . . Moreover, T [p] is non-negative so that ν 1,p (Ω) ≥ 0. Actually, it follows easily from the definition that KerT [p] is isomorphic to H p (Ω), the p-th absolute de Rham cohomology space of Ω with real coefficients. Therefore:

-a positive lower bound of ν 1,p (Ω) will imply in particular that H p (Ω) = 0; -a positive upper bound of ν 1,p (Ω) will be significant only when H p (Ω) = 0.

As Ω is connected, we see that H 0 (Ω) is 1-dimensional. Therefore, in our notation, ν 1,0 (Ω) = 0 and ν 2,0 (Ω) = ν 2 is the first positive eigenvalue of the classical problem [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF]. Finally, using the Hodge star operator, we define a dual operator T

[p] D , also acting on Λ p (Σ); in particular, the dual of T [n] defines an operator T

[0] D acting on C ∞ (Σ) and different from the classical Dirichlet-to-Neumann operator T (see Section 2.1 for details). The operator T [p] belongs to a family of operators depending on a complex parameter z, introduced by G. Carron in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] (see the proof of Theorem 11). Other Dirichlet to Neumann operators acting on differential forms, but different from T [p] , were introduced by Joshi and Lionheart in [START_REF] Joshi | An inverse boundary value problem for harmonic differential forms[END_REF], and Belishev and Sharafutdinov in [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF]. In the preprint [START_REF] Sharafutdinov | The complete Dirichlet-to-Neumann map for differential forms[END_REF], the operator ⋆ Σ T [p] : Λ p (Σ) → Λ n-p (Σ) appears in a certain matrix decomposition of the Joshi and Lionheart operator. None of these works, however, discuss eigenvalue estimates.

Lower bounds by the extrinsic geometry

First, some notations. Fix a point x ∈ Σ and let η 1 (x), . . . , η n (x) be the principal curvatures of Σ n at x (our sign convention is that the principal curvatures of the unit ball in R n+1 are positive). The p-curvatures of Σ are, by definition, all possible sums η j 1 (x) + • • • + η jp (x) for j 1 , . . . , j p ∈ {1, . . . , n}. Arrange the sequence of principal curvatures so that it is non-decreasing:

η 1 (x) ≤ • • • ≤ η n (x)
, and call

σ p (x) . = η 1 (x) + • • • + η p (x)
the lowest p-curvature at x. We say that Σ is p-convex if σ p (x) ≥ 0 for all x ∈ Σ, and let

σ p (Σ) = inf x∈Σ σ p (x).
Note that 1-convex means, simply, convex (all principal curvatures are non-negative) and n-convex means that Σ has non-negative mean curvature because, by definition, σ n (Σ) = nH, where H is a lower bound of the mean curvature of Σ. Finally, it is clear from the definition that, if Σ is p-convex, then it is q-convex for all q ≥ p.

Recall that, if ω is a p-form on Ω n+1 , the Bochner formula gives ∆ω = ∇ ⋆ ∇ω + W [p] ,

where W [p] is a symmetric endomorphism acting on Λ p (Ω), called the Bochner curvature term. One knows that W [1] = Ric, the Ricci tensor, hence W [1] ≥ 0 provided that Ω has nonnegative Ricci curvature.

From the work of Gallot and Meyer (see [START_REF] Gallot | Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne[END_REF]) we also know that, if γ is a lower bound of the eigenvalues of the Riemann curvature operator (seen as a symmetric endomorphism of Λ 2 (Ω)), then W [p] ≥ p(n + 1 -p)γ. Hence -if the curvature operator of Ω is nonnegative then W [p] ≥ 0 for all degrees p.

However, the condition W [p] ≥ 0 is sometimes much weaker than assuming the positivity of the curvature operator.

Theorem 1. Let p = 1, . . . , n. Assume that Ω n+1 satisfies W [p] ≥ 0 and that Σ is strictly p-convex, that is σ p (Σ) > 0. (a) If p < n + 1 2 then ν 1,p (Ω) > n -p + 2 n -p + 1 σ p (Σ).The equality never holds. (b) If p ≥ n + 1 2 then ν 1,p (Ω) ≥ p + 1 p σ p (Σ), (2)
which is an equality when Ω is a ball in the Euclidean space R n+1 .

Remark. Note that under the given curvature assumptions we have in particular H q (Ω) = 0 for all q ≥ p; so, the p-convexity has interesting topological consequences. This is not new: in [START_REF] Wu | Manifolds of partially positive curvature[END_REF] it was proved by other methods that, if σ p (Σ) > 0 and the sectional curvatures of Ω are non-negative, then Ω has the homotopy type of a CW -complex with cells only in dimensions ≤ p -1. For a result in negative curvature we refer to [START_REF] Savo | On the lowest eigenvalue of the Hodge Laplacian on compact, negatively curved domains[END_REF]: in particular, if Ω is a p-convex domain in H n then H p (Ω) = 0 for all q ≥ p, provided that p > (n + 1)/2. The proof of Theorem 1 uses a Reilly-type formula for differential forms, proved in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]. We characterize the equality in (2) in the following two cases: when p = n and when p > (n + 1)/ For Euclidean domains we also prove an inequality relating the first eigenvalues for consecutive degrees.

Theorem 4.

Let Ω be any compact domain in R n+1 , and let σ p (Σ) be a lower bound of the p-curvatures of Σ (which we do not assume to be positive).

(i) For all p = 1, . . . , n one has ν 1,p (Ω) ≥ ν 1,p-1 (Ω) + σ p (Σ)/p.

(ii) If Ω is convex, then ν 1,p > 0 for all p ≥ 1 and

ν 1,1 (Ω) ≤ ν 1,2 (Ω) ≤ • • • ≤ ν 1,n (Ω).
The inequality (i) is sharp for p > n+1 2 since equality is achieved by the unit Euclidean ball. The monotonicity property in (ii) is an immediate consequence of (i), because if Ω is convex then σ p (Σ) ≥ 0 for all p. We remark that the property (ii) holds also for the first eigenvalues of the Laplacian acting on p-forms of a convex Euclidean domain Ω, for the absolute boundary conditions (see [START_REF] Guerini | Eigenvalue and gap estimates for the Laplacian acting on p-forms[END_REF]).

Upper bounds by the isoperimetric ratio

It turns out that the existence of parallel forms implies that, for suitable degrees, the Dirichlet-to-Neumann eigenvalues can be bounded above by the isoperimetric ratio Vol(Σ)/Vol(Ω). Precisely, if Ω supports a non trivial parallel p-form, and H p (Ω) = H p R (Ω) = 0, then

ν 1,p-1 (Ω) + ν 1,n-p (Ω) ≤ Vol(Σ) Vol(Ω) . (3)
In some cases the estimate is sharp and we can characterize equality. Either one of the two cohomology assumptions can be removed if the given parallel form is known to be exact (respectively, co-exact): so, for example, (3) holds in all degrees for all domains in Euclidean space, since the parallel p-form dx 1 ∧ • • • ∧ dx p is exact and co-exact. The inequality (3) follows from the estimates in Section 4, which apply more generally to the ratio Σ ξ 2 / Ω ξ 2 , where ξ is a harmonic field, that is, a differential form which is closed and co-closed (we remark that on a manifold with nonempty boundary the vector space of harmonic fields of a given degree is infinite dimensional, and is properly contained in the space of harmonic forms).

As the volume form of Ω is parallel we have, for all compact manifolds with boundary, the estimate:

ν 1,n (Ω) ≤ Vol(Σ) Vol(Ω) , (4)
which reduces to an equality when Ω is a Euclidean ball. Then, we examine the equality case in (4). To that end, consider the mean-exit time function E, solution of the problem:

∆E = 1 on Ω, E = 0 on Σ.
Any domain for which the normal derivative ∂E/∂N is constant on Σ will be called a harmonic domain. The reason for this terminology is given by Proposition 18, in which we observe the following simple fact: ∂E/∂N is constant on Σ if and only if the mean value of any harmonic function on Ω equals its mean value on the boundary.

Theorem 5.

Let Ω be any compact domain. Then ν 1,n (Ω) ≤ Vol(Σ)/Vol(Ω). a) If equality holds, then Ω is a harmonic domain. b) Conversely, if Ω is a harmonic domain, then Vol(Σ)/Vol(Ω) belongs to the spectrum of T [n] (an associated eigenform being ⋆dE).

It remains to see how rigid the harmonicity condition is, and what conditions it imposes on the geometry of the boundary. For Euclidean domains the question was settled in a famous paper by Serrin [START_REF] Serrin | A symmetry problem in potential theory[END_REF] which states in particular that any harmonic domain in R n+1 is a ball. This rigidity result was extended by Kumaresan and Prajapat (see [START_REF] Kumaresan | Serrin's result for hyperbolic space and sphere[END_REF]) to domains in the hyperbolic space H n+1 and in the hemisphere S n+1 + . To our knowledge, the classification of harmonic domains in S n+1 is still an open (and interesting) question. Then, we have the following Corollary 6. a) For Euclidean domains the equality holds in (4) iff Ω is a ball. b) Let Ω be a domain in H n+1 or in S n+1

+ . If the equality holds in (4), then Ω is a geodesic ball.

Finally, using the estimate (4) and the inequalities of Theorem 1 and Theorem 4, one gets the following fact. Proposition 7. For the unit Euclidean ball B n+1 in R n+1 one has ν 1,p (B n+1 ) = p + 1 for all p ≥ (n + 1)/2. This calculation shows that the estimates of Theorems 1 and 4 are indeed sharp.

Remark. In a forthcoming paper, we will compute the whole spectrum of the Dirichletto-Neumann operator acting on p-forms of the unit Euclidean ball. In particular it turns out that, if 1 ≤ p < (n + 1)/2, then p + 1 is still an eigenvalue of T [p] , however it is no longer the first. In that range one has in fact ν 1,p (B n+1 ) = n + 3 n + 1 p.

Upper bounds by the Hodge-Laplace eigenvalues

The Hodge Laplacian acting on p-forms of a closed manifold Σ is the operator defined by

∆ Σ = d Σ δ Σ + δ Σ d Σ
, where d Σ and δ Σ denote respectively the differential and the co-differential acting on forms of Σ. We let λ ′ 1,p (Σ) (resp. λ ′′ 1,p (Σ)) be the first eigenvalue of ∆ Σ restricted to the subspace of exact (resp. co-exact) forms (these subspaces are preserved by ∆ Σ because it commutes with d Σ and δ Σ ). Differentiating eigenforms, one sees that, if

λ 1,p (Σ) is the first positive eigenvalue of ∆ Σ , then λ 1,p (Σ) = min{λ ′ 1,p (Σ), λ ′ 1,p+1 (
Σ)}. We then have the following lower bound.

Theorem 8. Assume that H p R (Ω) = 0, min(σ p (Σ), σ n-p+1 (Σ)) ≥ 0 and W [p] ≥ 0.
Then, for all p = 1, . . . , n:

λ ′ 1,p (Σ) ≥ 1 2 σ p (Σ)ν 1,n-p (Ω) + σ n-p+1 (Σ)ν 1,p-1 (Ω) .
Observe that λ ′ 1,1 (Σ) = λ 1 (Σ), the first positive eigenvalue of the Laplacian acting on functions of Σ. Taking p = 1 in the previous theorem we obtain the following sharp lower bound.

Theorem 9. Assume that Ω has non-negative Ricci curvature and that Σ is strictly convex, with principal curvatures bounded below by σ 1 (Σ) > 0. Then:

λ 1 (Σ) ≥ 1 2 σ 1 (Σ)ν 1,n-1 (Ω) + nHν 2,0 (Ω) ,
where H is a lower bound of the mean curvature of Σ, and ν 2,0 (Ω) is the first positive eigenvalue of the Dirichlet-to-Neumann operator on functions. Moreover, if n = dim(Σ) ≥ 3, the equality holds if and only if Ω is a Euclidean ball.

The motivation for looking at such a bound was given by the following estimate of Escobar [START_REF] Escobar | An isoperimetric inequality and the first Steklov eigenvalue[END_REF], which holds under the same assumptions of Theorem 9:

λ 1 (Σ) > nH 2 ν 2,0 (Ω). ( 5
)
We observe that the defect λ 1 (Σ) -nH 2 ν 2,0 (Ω) in ( 5) is bounded below by the first Dirichlet-to-Neumann eigenvalue in the degree n -1, thus obtaining a sharp bound.

An upper bound by the first biharmonic Steklov eigenvalue

The following problem on functions is classical, and is known as the fourth order (or biharmonic) Steklov eigenvalue problem:

   ∆ 2 f = 0 on Ω, f = 0, ∆f = µ ∂f ∂N on Σ. ( 6
)
For recent results on the problem, we refer to [START_REF] Ferrero | On a fourth order Steklov eigenvalue problem[END_REF] and [START_REF] Wang | Sharp bounds for the first non-zero Steklov eigenvalues[END_REF]. An immediate application of the min-max principle associated to the Dirichlet-to-Neumann operator on n-forms gives: [START_REF] Escobar | An isoperimetric inequality and the first Steklov eigenvalue[END_REF]. If the equality holds, then Ω is a harmonic domain.

Theorem 10. One has always µ 1 (Ω) ≥ ν 1,n (Ω), where µ 1 (Ω) is the first eigenvalue of (
In [START_REF] Wang | Sharp bounds for the first non-zero Steklov eigenvalues[END_REF] Wang and Xia prove that, if the Ricci curvature of Ω is non-negative and the mean curvature of Σ is bounded below by H > 0, then µ 1 (Ω) ≥ (n + 1)H. Moreover equality occurs if and only if Ω is isometric to a ball of R n+1 . Combining Theorem 10 and our estimate of Theorem 2 we see that, under the given assumptions:

µ 1 (Ω) ≥ ν 1,n (Ω) ≥ (n + 1)H
which implies the result of Wang and Xia. On the other hand, it is easy to observe that µ 1 (Ω) ≤ Vol(Σ)/Vol(Ω) (see for example [START_REF] Wang | Sharp bounds for the first non-zero Steklov eigenvalues[END_REF]). Then the estimate ( 4) is a direct consequence of this fact and Theorem 10.

The paper is organized as follows. In Section 2 we state the main properties of the operator T [p] . In Section 3 we prove the lower bounds and in Section 4 we give the proof of the upper bounds. Finally, in Section 5, we prove a rigidity result needed for the equality case of Theorem 3.

Generalities on the Dirichlet-to-Neumann operator

Before stating the main properties of T [p] , let us recall the following well-known facts. The Hodge-de Rham theorem for manifolds with boundary asserts that H p dR (Ω, R), the absolute de Rham cohomology space in degree p with real coefficients, is isomorphic to the (finite dimensional) vector space of harmonic p-forms φ satisfying the absolute boundary conditions (i N φ = i N dφ = 0 on Σ), which we denote by H p (Ω). Equivalently, one has:

H p (Ω) = {φ ∈ Λ p (Ω) : dφ = δφ = 0 on Ω, i N φ = 0 on Σ}.
By duality, the relative de Rham cohomology space in degree p is isomorphic to the vector space

H p R (Ω) = {φ ∈ Λ p (Ω) : dφ = δφ = 0 on Ω, J ⋆ φ = 0 on Σ}. Theorem 11.
Let Ω n+1 be a compact domain with smooth boundary Σ n . Let T [p] be the Dirichlet-to-Neumann operator acting on p-forms of Σ, as defined in Section 1.1. Then:

(a) T [p] is nonnegative and self-adjoint.

(b) The kernel of T [p] consists of the boundary values of absolute cohomology classes, and the restriction J ⋆ induces an isomorphism between H p (Ω) and Ker(T [p] ).

(c) T [p] is an elliptic pseudo-differential operator of order one. Hence it admits an increasing sequence of eigenvalues with finite multiplicities

ν 1,p (Ω) ≤ ν 2,p (Ω) ≤ . . . with ν 1,p (Ω) = 0 repeated b p (Ω) = dim H p (Ω) times. In particular, ν 1,p (Ω) > 0 if and only if H p (Ω) = 0.
(d) The first eigenvalue of T [p] satisfies the min-max principle

ν 1,p (Ω) = inf Ω d φ 2 + δ φ 2 Σ φ 2 (7)
where the infimum is taken over all p-forms φ on Ω such that i N φ = 0 on Σ.

We remark that (b) has already been observed in [START_REF] Sharafutdinov | The complete Dirichlet-to-Neumann map for differential forms[END_REF].

Proof. (a) We prove that the operator is self-adjoint. Recall the Stokes formula:

Ω dω 1 , ω 2 = Ω ω 1 , δω 2 - Σ J ⋆ ω 1 , i N ω 2
for all ω 1 ∈ Λ p-1 (Ω) and ω 2 ∈ Λ p (Ω). Now let φ, ψ ∈ Λ p (Σ) and denote by φ, ψ their harmonic tangential extensions on Ω. The definition of T [p] and the Stokes formula give:

Σ T [p] φ, ψ = - Σ i N d φ, J ⋆ ψ = Ω d φ, d ψ -δd φ, ψ .
As ψ is harmonic and i N ψ = 0 we have

- Ω δd φ, ψ = Ω dδ φ, ψ = Ω δ φ, δ ψ . So Σ T [p] φ, ψ = Ω d φ, d ψ + δ φ, δ ψ which shows that T [p] is self-adjoint. Taking ψ = φ yields: Σ T [p] φ, φ = Ω d φ 2 + δ φ 2 ≥ 0.
and

T [p] is nonnegative. (b) If φ ∈ Ker(T [p]
) then its harmonic tangential extension φ satisfies, on Σ:

i N φ = i N d φ = 0.
Hence φ is the restriction of a form (cohomology class) in H p (Ω). Conversely, it is clear by the definition that an absolute cohomology class restricts to a form in the kernel of T [p] . Then:

Ker(T [p] ) = J ⋆ (H p (Ω)).
We observe that the map J ⋆ : H p (Ω) → J ⋆ H p (Ω) is injective: in fact, if J ⋆ φ = 0 for some cohomology class φ, then φ is harmonic and zero on the boundary, which implies φ = 0. Then the dimension of Ker(T [p] ) equals b p (Ω).

(c) The proof that T [p] is an elliptic pseudo-differential operator follows the lines of the proof done in Section 6.4 of [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF]. There, in studying determinants, G. Carron considers the linear operator

T z : Λ p (Σ) → Λ p (Σ) depending on a complex parameter z ∈ C \ [0, ∞)
and defined by

T z φ = -i N d φz ,
where φz is the unique solution of ∆ φz = z φz on Ω,

i N φz = 0, J ⋆ φz = φ on Σ. ( 8
)
Carron shows that T z is an elliptic, pseudo-differential, invertible operator. In fact, the inverse S z of T z is shown to be the operator obtained by restricting to the boundary the Green kernel of the Hodge Laplacian ∆ acting on p-forms of Ω, for the absolute boundary conditions; as S z is pseudo-differential of order -1, the operator T z is pseudo-differential of order 1. The restriction on z is imposed precisely because then T z will be invertible, since z avoids the spectrum of ∆ (which is contained in the nonnegative half-line).

Our operator is obtained by taking z = 0 in (8): it is no longer invertible when H p (Ω) = {0} but it is still pseudo-differential and elliptic because, by (b), its kernel is finite dimensional, isomorphic to H p (Ω). In fact, the operator S 0 is now invertible modulo compact operators, given by the projection onto the kernel of T 0 and its transpose. The rest of Carron's proof carries over and so T 0 = T [p] is an elliptic PDO of order 1. More generally, T z is an elliptic PDO for all z, and is invertible as long as z does not belong to the spectrum of ∆. The rest of (c) now follows from the standard theory of elliptic PDO (see [START_REF] Shubin | Pseudodifferential Operators and Spectral Theory[END_REF]).

(d) The min-max principle gives

ν 1,p (Ω) = inf Ω d φ 2 + δ φ 2 Σ φ 2 : ∆ φ = 0, i N φ = 0 .
We only have to show that we can remove the condition ∆ φ = 0. This follows from the fact that among all tangential extensions ξ of a given form φ ∈ Λ p (Σ), the harmonic tangential extension φ minimizes the quadratic form Ω dξ 2 + δξ 2 . Indeed, assume that J ⋆ ξ = φ = J ⋆ φ and i N ξ = 0 = i N φ. Let ψ = ξ -φ so that ψ = 0 on the boundary.

Using the Stokes formula one verifies that:

0 ≤ Ω dψ 2 + δψ 2 = Ω dξ 2 + δξ 2 - Ω d φ 2 + δ φ 2 ,
and the assertion follows.

The dual problem

Let p = 0, . . . , n. Given a p-form φ on Σ consider the unique (p + 1)-form φ on Ω which satisfies: ∆ φ = 0 on Ω

J ⋆ φ = 0, i N φ = φ on Σ.
The form φ will be called the harmonic normal extension of φ. Its existence and uniqueness is also proved in Schwarz [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF]. We set

T [p] D φ = J ⋆ (δ φ)
and call T

ν D 1,p (Ω) = ν 1,n-p (Ω).
Moreover, the min-max principle for the dual problem takes the form:

ν D 1,p (Ω) = inf Ω d φ 2 + δ φ 2 Σ φ 2 : φ ∈ Λ p+1 (Ω), J ⋆ φ = 0 . (9)
Note that T

[0]

D is an operator acting on functions, which clearly differs from the operator

T [0] .
3 Lower bounds: proofs

Reilly formula for differential forms

The main tool used in the proof of the lower bound is a Reilly-type formula for differential forms proved by the authors in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF], which we state below. Denote by S the shape operator of the immersion of Σ in Ω; it is defined as S(X) = -∇ X N for all tangent vectors X ∈ T Σ. S admits a canonical extension acting on p-forms on Σ and denoted by S [p] . Explicitly, if ω is a p-form on Σ one has:

S [p] ω(X 1 , . . . , X p ) = p j=1 ω(X 1 , . . . , S(X j ), . . . , X p ),
for tangent vectors X 1 , . . . , X p ∈ T Σ. It is clear from the definition that the eigenvalues of S [p] are precisely the p-curvatures of Σ: therefore we have immediately

S [p] ω, ω ≥ σ p (Σ) ω 2
at all points of Σ and for all p-forms ω. Now let ω be a p-form on Ω. The Reilly formula says that [START_REF] Joshi | An inverse boundary value problem for harmonic differential forms[END_REF] where the boundary term has the following expression:

Ω dω 2 + δω 2 = Ω ∇ω 2 + W [p] (ω), ω + 2 Σ i N ω, δ Σ (J ⋆ ω) + Σ B(ω, ω),
B(ω, ω) = S [p] (J ⋆ ω), J ⋆ ω + nH i N ω 2 -S [p-1] (i N ω), i N ω = S [p] (J ⋆ ω), J ⋆ ω + S [n-p+1] (J ⋆ ⋆ ω), J ⋆ ⋆ ω
By convention, we set S [0] = S [n+1] = 0. For a detailed proof of (10) see [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF].

Proof of Theorem 1

We assume that W [p] ≥ 0, and that the p-curvatures of Σ are bounded below by σ p (Σ) > 0. We have to prove that, if p < n+1 2 then:

ν 1,p (Ω) > n -p + 2 n -p + 1 σ p (Σ), (11)
and if p ≥ n+1 2 then ν 1,p (Ω) ≥ p + 1 p σ p (Σ). ( 12
)
Let ω be an eigenform associated to ν 1,p (Ω) and let ω be its harmonic tangential extension to Ω. By the variational characterization [START_REF] Ferrero | On a fourth order Steklov eigenvalue problem[END_REF]:

Ω dω 2 + δ ω 2 = ν 1,p (Ω) Σ ω 2 (13)
because, on the boundary, ω 2 = ω 2 + i N ω 2 = ω 2 . We apply the Reilly formula to ω. As W [p] ≥ 0 and i N ω = 0 we get

Ω dω 2 + δ ω 2 ≥ Ω ∇ω 2 + Σ S [p] (ω), ω ≥ Ω ∇ω 2 + σ p (Σ) Σ ω 2 (14)
We will use the following estimate of Gallot and Meyer [START_REF] Gallot | Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne[END_REF], valid for any p-form ω:

∇ω 2 ≥ dω 2 p + 1 + δ ω 2 n -p + 2 . ( 15
)
When p < n+1 2 one has p + 1 < n -p + 2 hence:

∇ω 2 ≥ dω 2 + δ ω 2 n -p + 2 , (16)
and the equality implies dω = 0. Inserting ( 16) in ( 14), and taking into account [START_REF] Ros | Compact Hypersurfaces with constant higher order mean curvatures[END_REF], we obtain [START_REF] Kumaresan | Serrin's result for hyperbolic space and sphere[END_REF]. Note that then ν 1,p (Ω) > 0. Equality in [START_REF] Serrin | A symmetry problem in potential theory[END_REF] implies that dω = 0 hence i N dω = 0: but this is impossible because otherwise ν 1,p (Ω) = 0. So the inequality is always strict. If p ≥ n+1 2 one has ∇ω 2 ≥ dω 2 + δ ω 2 p + 1 [START_REF] Sharafutdinov | The complete Dirichlet-to-Neumann map for differential forms[END_REF] and proceeding as before we obtain [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]. The inequality ( 12) is sharp: for the unit Euclidean ball we have σ p (Σ) = p and ν 1,p (B n+1 ) = p + 1 (see Proposition 7). We finally remark that, if p > n+1 2 and the equality holds in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF], it holds also in [START_REF] Sharafutdinov | The complete Dirichlet-to-Neumann map for differential forms[END_REF] and then δ ω = 0. Now we study the equality case of this estimate. Recall that the p-form ω is a conformal Killing form if it satisfies the differential equation

∇ X ω = 1 p + 1 i X dω - 1 n -p + 2 X * ∧ δ ω
for all X ∈ T Ω. A co-closed conformal Killing form is called a Killing form. It is wellknown that the inequality ( 15) is an equality if and only if ω is a conformal Killing form (see for example [START_REF] Gallot | Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne[END_REF]). We then have:

Proposition 12. Assume p ≥ (n + 1)/2. If equality holds in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF] then the harmonic tangential extension of a p-eigenform associated to ν 1,p (Ω) is a conformal Killing p-form (a Killing form if p > n+1 2 ) and the p lowest principal curvatures of the boundary are constant, equal to c = ν 1,p (Ω)/(p + 1).

Proof. Looking at the proof of [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF] we see immediately that if the equality holds then ω is a conformal Killing form and, by the last remark in the proof, it is a Killing form when p > n+1 2 . It remains to show the last assertion. Now, the Gauss formula leads to the following relations (see Section 6 in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]):

∇ Σ X (i N ω) = i N ∇ X ω -i S(X) J ⋆ ω ∇ Σ X (J ⋆ ω) = J ⋆ (∇ X ω) + S(X) ⋆ ∧ i N ω, (18)
for all X ∈ T Σ, where ∇ Σ is the Levi-Civita connection of Σ. Since ω is the harmonic tangential extension of ω, we have i N ω = 0 and the first equation in ( 18) reads:

i N ∇ X ω = i S(X) ω. (19)
On the other hand, since ω is a conformal Killing p-form we have for all X ∈ Γ(T Σ):

i N ∇ X ω = - 1 p + 1 i X (i N dω) = ν 1,p (Ω) p + 1 i X ω. (20)
We used the fact that i N δ ω = -δ Σ (i N ω) = 0, which immediately implies i N (X ⋆ ∧δ ω) = 0. Combining ( 19) and ( 20) gives:

i S(X)- ν 1,p (Ω) p+1 X ω = 0 (21)
for all X ∈ Γ(T Σ). The form ω, being an eigenform of an elliptic operator, can't vanish on an open set and therefore is non-zero a.e. on Σ. Take a point x where it does not vanish: then, at x, there exists p principal directions, say v 1 , . . . , v p , such that ω(v 1 , . . . , v p ) = 0. Choosing successively X = v 1 , . . . , v p one sees from (21) that the associated principal curvatures satisfy

λ 1 = • • • = λ p = ν 1,p (Ω)
p+1 .

Proof of Theorem 2

Assume that Ω n+1 has nonnegative Ricci curvature and that Σ has mean curvature bounded below by H > 0. Then σ n = nH and applying Theorem 1 for p = n we get ν 1,n (Ω) ≥ (n + 1)H. It remains to show that, if the equality holds, then Ω is a Euclidean ball. Now, under the given assumptions, we have Vol(Σ)/Vol(Ω) ≥ (n + 1)H by Theorem 1 in [START_REF] Ros | Compact Hypersurfaces with constant higher order mean curvatures[END_REF], with equality if and only if Ω is a Euclidean ball. It is then enough to show that Vol(Σ) Vol(Ω) = (n + 1)H.

From Proposition 12, we know that if ω ∈ Λ n (Σ) is a eigenform associated with ν 1,n (Ω) = (n + 1)H, then its harmonic tangential extension ω ∈ Λ n (Ω) is a Killing n-form on Ω; in particular, δ ω = 0. We can write dω = f Ψ Ω , where Ψ Ω is the volume form of Ω and f is a smooth function. As ω is harmonic and co-closed, we have

0 = δdω = δ(f Ψ Ω ) = -i ∇f Ψ Ω ,
which immediately implies ∇f = 0. By renormalization, we can assume that f = 1 and so dω is the volume form of Ω. By assumption, J ⋆ dω = 0 and i N dω = -(n + 1)Hω. Then, on Σ

1 = i N dω 2 = (n + 1) 2 H 2 ω 2 .
On the other hand, by the Stokes formula and the fact that dω has constant unit norm:

Vol(Ω) = Ω dω 2 = - Σ ω, i N dω = (n + 1)H Σ ω 2 = Vol(Σ) (n + 1)H ,
which proves the assertion.

The equality case for Euclidean domains: proof of Theorem 3

We fix c > 0 and let F p (c) denote the set of p-forms ω on Ω, p = 0, . . . , n, with the following properties: a) ω is harmonic and tangential (that is i N ω = 0 on Σ). b) ω is Killing and dω is parallel. c) i N dω = -(p + 1)cω, where ω = J ⋆ ω is the restriction to Σ.

Note that F 0 (c) consists of all harmonic functions f with parallel gradient and such that Proof. The Cartan formula gives di V ω + i V dω = L V ω, where L V is the Lie derivative along V . If V is parallel and ω is Killing, we have L V ω = ∇ V ω = 1 p+1 i V dω and then:

∂ f ∂N = -c f : if f is not trivial, its
di V ω = - p p + 1 i V dω. ( 22
)
Now ∇ V dω = L V dω = di V dω = 0 by Cartan formula and ( 22). This holds for all parallel vector fields: in particular, any Killing form of degree p ≥ 1 in Euclidean space has parallel exterior derivative.

Fix ω ∈ F p (c). As V is parallel, i V commutes with ∆ and anticommutes with i N . Then i V ω satisfies a). As i V anticommutes with δ, we see that i V ω is co-closed. On the other hand, since V is parallel:

∇ X i V ω = i V ∇ X ω = 1 p + 1 i V i X dω = - 1 p + 1 i X i V dω = 1 p i X di V ω
where we used (22) in the last equality. Hence i V ω is a Killing (p -1)-form. A similar calculation shows that ∇ X di V ω = 0, hence di V ω is parallel and b) follows.

Finally, again using (22): andc) follows as well.

i N di V ω = - p p + 1 i N i V dω = p p + 1 i V i N dω = -pci V ω,
Now assume that Ω is an extremal domain for our inequality, and let ω be the tangential harmonic extension of an eigenform ω associated to ν 1,p (Ω). Set c = ν 1,p (Ω)/(p + 1). By Proposition 12, ω is a Killing p-form: in particular, as observed in the proof of the Lemma 13, dω is parallel. Moreover, i N dω = -(p + 1)cω by definition. This means that ω is a form in F p (c). As ω is non trivial, we can find p parallel vector fields V 1 , . . . , V p such that the function f = ω(V 1 , . . . , V p ) is non trivial. Applying the lemma successively to the parallel fields V 1 , . . . , V p , we see that f ∈ F 0 (c), that is, f satisfies

   ∇d f = 0 on Ω, ∂ f ∂N = -c f on Σ.
By Proposition 12, the lowest p principal curvatures are constant, equal to c, and then S ≥ c. We now apply Theorem 19 in the Appendix, to conclude that Ω is a Euclidean ball. The proof of Theorem 3 is now complete.

An inequality for consecutive degrees: proof of Theorem 4

We have to show that if Ω is a domain in R n+1 , then for all p = 1, . . . , n:

ν 1,p (Ω) ≥ ν 1,p-1 (Ω) + σ p (Σ) p . ( 23
)
For the proof, we consider the family of unit length parallel vector fields on R n+1 , which is naturally identified with S n . Let ω ∈ Λ p (Σ) be an eigenform associated to the eigenvalue ν 1,p (Ω) and denote by ω its harmonic tangential extension. Let V be a unit length parallel vector field. Since ∆ commutes with the contraction i V , the (p -1)-form i V ω is harmonic. Moreover we clearly have i N i V ω = 0. Hence we can use i V ω as test form for the eigenvalue ν 1,p-1 (Ω), and by the min-max principle we have

ν 1,p-1 (Ω) Σ i V ω 2 ≤ Ω di V ω 2 + δi V ω 2 (24)
for all V ∈ S n . Now we want to integrate this inequality with respect to V ∈ S n . In order to simplify the formulae, we use the renormalized measure

dµ = n + 1 Vol(S n ) dvol S n ,
where dvol S n is the canonical measure of S n . Then, we have the following identities, which are valid pointwise and are proved in [START_REF] Guerini | Eigenvalue and gap estimates for the Laplacian acting on p-forms[END_REF] (Lemma 4.8, p. 336):

S n i V ω 2 dµ(V ) = p ω 2 S n di V ω 2 dµ(V ) = ∇ω 2 + (p -1) dω 2 S n δi V ω 2 dµ(V ) = S n i V δ ω 2 dµ(V ) = (p -1) δ ω 2 .
Integrating (24) with respect to V ∈ S n and using the previous identities, we then have, by the Fubini theorem:

pν 1,p-1 (Ω) Σ ω 2 ≤ Ω ∇ω 2 + (p -1) Ω dω 2 + δ ω 2 .
On the other hand, the Reilly formula (10) applied to ω gives:

Ω dω 2 + δ ω 2 = Ω ∇ω 2 + Σ S [p] (J ⋆ ω, )J ⋆ ω ≥ Ω ∇ω 2 + σ p (Σ) Σ ω 2 .
Eliminating Ω ∇ω 2 in the previous two inequalities leads to:

pν 1,p-1 (Ω) Σ ω 2 ≤ p Ω dω 2 + δ ω 2 -σ p (Σ) Σ ω 2 = pν 1,p (Ω) Σ ω 2 -σ p (Σ) Σ ω 2
Dividing both sides by p Σ ω 2 proves (23).

4 Upper Bounds: proofs

Upper bounds by the isoperimetric ratio

A p-form ξ is said to be a harmonic field if dξ = δξ = 0. We start from the following: Proposition 14. Let ξ be a harmonic field of degree p on Ω.

(a) If ξ is exact and p = 2, . . . , n

+ 1 then ν 1,p-1 (Ω) Ω ξ 2 ≤ Σ i N ξ 2 . (b) If ξ is exact and p = 1 then ν 2,0 (Ω) Ω ξ 2 ≤ Σ i N ξ 2 . (c) If ξ is co-exact and p = 1, . . . , n then ν 1,n-p (Ω) Ω ξ 2 ≤ Σ J ⋆ ξ 2 .
Proof. (a) By the Hodge-Morrey decomposition (see [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF]) if ξ is an exact p-form, there is a unique co-exact (hence co-closed) (p -1)-form ω, called the canonical primitive of ξ, which satisfies:

dω = ξ i N ω = 0 on Σ.
We use ω as a test-form for the eigenvalue ν 1,p-1 (Ω) and then

ν 1,p-1 (Ω) Σ ω 2 ≤ Ω dω 2 .
By the Stokes formula Ω dω 2 = -Σ i N dω, J ⋆ ω ; by the Schwarz inequality

Ω dω 2 2 ≤ Σ i N dω 2 • Σ ω 2 .
Eliminating Σ ω 2 from the previous two inequalities we get

ν 1,p-1 (Ω) Ω dω 2 ≤ Σ i N dω 2 ,
which is the assertion. We remark that the equality holds if and only if the canonical primitive of ξ is an eigenform of T [p-1] associated to ν 1,p-1 (Ω). (b) If ξ is an exact harmonic field of degree 1, then ξ = df for an harmonic function f . We can assume that f integrates to zero on Σ, and so we can use f as a test function for the eigenvalue ν 2,0 (Ω). The rest of the proof is as in (a). (c) Let ξ be a co-exact p-harmonic field. Then ⋆ξ is an exact (n -p + 1)-harmonic field and we can apply (a) to it. The inequality follows because

i N ⋆ ξ 2 = ⋆ Σ J ⋆ ξ 2 = J ⋆ ξ 2 .
If the equality holds, then the canonical primitive of ⋆ξ is an eigenform of T [n-p] associated to ν 1,n-p (Ω). We can also characterize the equality by duality, as follows. If ξ is co-exact, it has a unique canonical co-primitive, that is, a unique exact (p + 1)-form α such that:

δα = ξ J ⋆ α = 0 on Σ.
It is clear that if we have equality then α is an eigenform of the dual operator

T [p] D associated to ν D 1,p (Ω) = ν 1,n-p (Ω). That is, J ⋆ (δα) = ν 1,n-p (Ω)i N α.
We remark that if H p (Ω) = 0 (resp. H p R (Ω) = 0) then any p-harmonic field is automatically exact (resp. co-exact). Therefore, as at any point of the boundary one has ξ 2 = J ⋆ ξ 2 + i N ξ 2 , we have, summing the two inequalities of the Proposition:

Corollary 15. Assume that H p (Ω) = H p R (Ω) = 0. Let ξ be a harmonic field of degree p. (a) If p = 2, . . . , n then ν 1,p-1 (Ω) + ν 1,n-p (Ω) ≤ Σ ξ 2 / Ω ξ 2 . If ξ is parallel then it has constant norm and ν 1,p-1 (Ω) + ν 1,n-p (Ω) ≤ Vol(Σ) Vol(Ω) . (b) If p = 1 then ν 2,0 (Ω) + ν 1,n-1 (Ω) ≤ Σ ξ 2 / Ω ξ 2 .
(c) In particular, if H 1 R (Ω) = 0 and f is any harmonic function then

ν 2,0 (Ω) + ν 1,n-1 (Ω) ≤ Σ df 2 Ω df 2 .
On the other hand, the volume form of Ω is parallel, exact and has degree n + 1. Then it follows directly from the first point of the Proposition 14 that, for all compact manifolds with boundary, one has the sharp bound:

ν 1,n (Ω) ≤ Vol(Σ) Vol(Ω) . ( 25
)
We have equality in (25) when Ω = B n+1 is the unit Euclidean ball: in fact Vol(Σ)/Vol(Ω) = n + 1 and by the main lower bound (Theorem 2) we have ν 1,n (B n+1 ) ≥ n + 1. So ν 1,n (B n+1 ) = n + 1. We will reprove (25) and discuss its equality case in Section 4.2.

We end this section with the following calculation.

Proposition 16. We have ν 1,p (B n+1 ) = p + 1 for all p ≥ n+1 2 . Proof. Let Ω = B n+1 and let ν 1,p = ν 1,p (B n+1 ). We just observed that ν 1,n = n + 1. We now use Theorem 4; as σ p (Σ)/p = 1 for all p, we see that ν 1,p ≥ ν 1,p-1 +1. Then ν 1,n-1 ≤ n and, by induction, ν 1,p ≤ p + 1 for all p. However, when p ≥ (n + 1)/2, Theorem 1 applied to Ω gives ν 1,p ≥ p + 1 and so ν 1,p = p + 1.

For later use, we observe the following Proposition 17. Assume that Ω supports a non constant linear function, that is, a smooth function f with df non trivial and parallel. If H 1 R (Ω) = 0, then:

ν 2,0 (Ω) + ν 1,n-1 (Ω) ≤ Vol(Σ) Vol(Ω) .
(Ω) = ν 1,n-1 (Ω): J ⋆ (δα) = ν 1,n-1 (Ω)i N α. As J ⋆ (δα) = d Σ f we have d Σ f = ν 1,n-1 (Ω)i N α. It follows that ∆ Σ f = δ Σ d Σ f = ν 1,n-1 (Ω)δ Σ i N α = -ν 1,n-1 (Ω)i N δα = -ν 1,n-1 (Ω) ∂f ∂N = ν 2,0 (Ω)ν 1,n-1 (Ω)f,
that is, f is an eigenfunction of ∆ Σ associated to ν 2,0 (Ω)ν 1,n-1 (Ω), as asserted. Observe that then ν 1,n-1 (Ω) > 0 otherwise f would be constant. To prove the first assertion, recall that, for any smooth function on Ω one has, at all points of Σ:

∆f = ∆ Σ f - ∂ 2 f ∂N 2 + nH ∂f ∂N .
As ∇ 2 f = 0, we have ∆f = 0 and ∂ 2 f ∂N 2 = 0, and we easily obtain nH = ν 1,n-1 (Ω). (b) The equality holds for the Euclidean unit ball, by Proposition 7 (it is known that ν 2,0 (B n+1 ) = 1). Now, if the equality holds, then Σ has constant mean curvature by (a), hence Σ is a sphere by a well-known result of Alexandrov.

Harmonic domains

Recall that the domain Ω is called harmonic if ∂E/∂N is constant on Σ, where E is the mean-exit time function, solution of the problem ∆E = 1 on Ω, E = 0 on Σ. Any ball in a constant curvature space form is harmonic, simply because the mean-exit time function is radially symmetric. We observe the following equivalent condition.

Proposition 18. Ω is harmonic if and only if, for all harmonic functions f on Ω, one has:

1 Vol(Ω) Ω f = 1 Vol(Σ) Σ f
(that is, the mean value of any harmonic function on the domain equals its mean value on the boundary).

Proof. Assume that Ω is harmonic and let f be any harmonic function on Ω. By the definition of E and the Green formula, we have:

Ω f = Ω f ∆E = Σ f ∂E ∂N .
As ∂E/∂N is constant, say equal to c, we have Ω f = c Σ f . Taking f = 1 we see that c = Vol(Ω)/Vol(Σ) and the first half is proved. Conversely, assume that the above mean-value property is true for all harmonic functions on Ω . Fix a point x ∈ Σ and let f k ∈ C ∞ (Σ) be a sequence of functions converging to the Dirac measure of Σ at x as k → ∞. Let fk be the harmonic extension of f k .

Then

Ω fk = Σ f k ∂E/∂N and the assumption gives Vol(Ω) Vol(Σ) Σ f k = Σ f k ∂E ∂N for all k. Letting k → ∞ we obtain Vol(Ω) Vol(Σ) = ∂E ∂N (x).
As x is arbitrary, we see that ∂E/∂N is indeed constant on Σ.

Proof of Theorem 5

It is perhaps simpler to reprove the inequality using the dual operator T By the Schwarz inequality:

Σ ∂E ∂N 2 ≥ 1 Vol(Σ) Σ ∂E ∂N 2 = Vol(Ω) 2 Vol(Σ)
and the inequality follows immediately. If the equality holds then ∂E/∂N must be constant and then Ω is a harmonic domain. Conversely, assume that Ω is harmonic. Then the normal derivative of E is constant along Σ, and equals c = Vol(Ω)/Vol(Σ). Let α = dE. Then

   ∆α = 0 J ⋆ α = 0, i N α = ∂E ∂N = c.
By the definition of T Fix a degree p = 1, . . . , n. We assume that H p R (Ω) = 0, min(σ p (Σ), σ n-p+1 (Σ)) ≥ 0 and W [p] ≥ 0. We have to show:

λ ′ 1,p (Σ) ≥ 1 2 σ p (Σ)ν 1,n-p (Ω) + σ n-p+1 (Σ)ν 1,p-1 (Ω) . (26)
Let φ be a co-exact eigenform associated to λ = λ ′′ 1,p-1 (Σ) = λ ′ 1,p (Σ) and consider the exact p-eigenform ω = d Σ φ also associated to λ. Let φ be a solution of ∆ φ = 0 on Ω, J ⋆ φ = φ, J ⋆ (δ φ) = 0 on Σ, which exists by Lemma 3.4.7 in [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF]. Then, using the Stokes formula one checks that δd φ = 0 on Ω (the extension φ first appeared in the paper of Duff and Spencer [START_REF] Duff | Harmonic tensors on Riemannian manifolds with boundary[END_REF]). If we let ω = d φ, then ω is an exact p-harmonic field satisfying:

dω = δ ω = 0 on Ω J ⋆ ω = ω on Σ.
We apply the Reilly formula [START_REF] Joshi | An inverse boundary value problem for harmonic differential forms[END_REF] to ω; as

W [p] ≥ 0 and δ Σ (J ⋆ ω) = δ Σ d Σ φ = λφ we obtain -2λ Σ i N ω, φ ≥ Σ S [p] (J ⋆ ω), J ⋆ ω + S [n-p+1] (J ⋆ ⋆ ω), J ⋆ ⋆ ω .
The Stokes formula gives:

Σ i N ω, φ = Σ i N d φ, J ⋆ φ = Ω φ, δd φ - Ω d φ 2 = - Ω ω 2 .
By our curvature assumptions, we end-up with 2λ

Ω ω 2 ≥ σ p (Σ) Σ J ⋆ ω 2 + σ n-p+1 (Σ) Σ i N ω 2 .
The p-harmonic field ω is exact, and also co-exact because H p R (Ω) = 0. We can then apply Proposition 14 to estimate the boundary integrals in the right hand side, and the estimate (26) follows.

Proof of Theorem 9

Let λ 1 (Σ) be the first positive eigenvalue of the Laplacian on functions of Σ. We assume that Ω has nonnegative Ricci curvature and that Σ is strictly convex, with principal curvatures bounded below by σ 1 (Σ) > 0. We have to show that

λ 1 (Σ) ≥ 1 2 (σ 1 (Σ)ν 1,n-1 (Ω) + nHν 2,0 (Ω)) . (27)
Moreover, if n = dim(Σ) ≥ 3, the equality holds if and only if Ω is a Euclidean ball.

Let f be a first eigenfunction associated to µ 1 (Ω). As J ⋆ (df ) = 0 we can use df as a test-form in [START_REF] Guerini | Eigenvalue and gap estimates for the Laplacian acting on p-forms[END_REF]. Then

ν D 1,0 (Ω) ≤ Ω (∆f ) 2 Σ ∂f ∂N 2 = µ 1 (Ω)
where the equality follows from the Rayleigh-Ritz characterization of µ 1 (Ω) (see [START_REF] Ferrero | On a fourth order Steklov eigenvalue problem[END_REF]). If equality holds, then df must be an eigenform of T

[0]

D associated to ν D 1,0 (Ω), hence ∆df = 0. But then ∆f is a constant, and we can assume ∆f = 1. As f = 0 on Σ we see that f = E, the mean-exit time function, and the boundary conditions satisfied by f imply that the normal derivative of E is constant. Hence Ω is harmonic.

Appendix

Here we state a general result which gives sufficient conditions on a manifold to be isometric with a Euclidean ball. This result is used in the proof of Theorem 3 but it is perhaps of independent interest. If Ω has non-negative sectional curvature and the second fundamental form of Σ satisfies S ≥ c, then Ω is isometric with a Euclidean ball.

Proof. It is enough to prove that the boundary is isometric to a round sphere. Then, by Theorem 1 in [START_REF] Xia | Rigidity of compact manifolds with boundary and nonnegative Ricci curvature[END_REF], we conclude that (Ω n+1 , g) is isometric with a Euclidean ball.

Here are the main steps. We prove that: a) Σ is connected. b) Ric Σ ≥ (n -1)c 2 . c) Σ has diameter greater than or equal to π c . The proof of the Theorem will follow by observing that, by Myers' theorem and a), b), one has diam(Σ) ≤ π c ; hence, by c), the diameter is equal to π c . By the rigidity theorem of Cheng [START_REF] Cheng | Eigenvalue comparison theorems and its geometric applications[END_REF], Σ is isometric to a sphere of radius 1/c, as asserted. We prove a). Looking at the long exact sequence of the pair (Ω, Σ), it is enough to show that H 1 R (Ω) = 0: in fact, in that case H 0 (Σ) ∼ H 0 (Ω) ∼ R. Now the Ricci curvature of Ω is non-negative and the mean curvature of Σ is bounded below by c > 0: by Theorem 2 we have ν 1,n (Ω) > 0 and then H n (Ω) = H 1 R (Ω) = 0.

We prove b). It is enough to prove that, for any unit length tangent vector X ∈ T Σ, one has Ric Σ (X, X) ≥ (n -1)c 2 . The Gauss lemma and the non-negativity of the sectional curvatures of Ω give:

Ric Σ (X, X) ≥ nH S(X), X -|S(X)| 2 .

Fix an orthonormal frame (e 1 , . . . , e n ) of principal directions, so that S(e j ) = η j e j for all j. Then:

Ric Σ (X, X) ≥ n j=1 (η j (nH -η j )) X, e j 2 ; as η j ≥ c for all j one sees that η j (nH -η j ) ≥ (n -1)c 2 for all j and the assertion follows.

We finally prove c). Since ∇f is parallel we have that |∇f | is constant on Ω, and we can assume that it is equal to 1. The restriction of f is continuous on Σ, which is compact: then let p + ∈ Σ (resp. p -∈ Σ) be a point where the restriction of f is maximum (resp. minimum). We prove d(p -, p + ) ≥ π c . Now:

1 = |∇f | 2 (p ± ) = |∇ Σ f | 2 (p ± ) + ∂f ∂N 2 (p ± ) = c 2 f (p ± ) 2 .
The function f is not constant on Σ (because it is harmonic on Ω and c > 0) therefore:

f (p + ) = 1 c , f (p -) = - 1 c .
As Σ is connected, there exists a minimizing geodesic γ : [0, l] → Σ parametrized by arc length and joining p -with p + . So we have γ(0) = p -, γ(l) = p + and the distance from p -to p + is l. It is now enough to prove that l ≥ π c . Let α(t) := f • γ(t) for t ∈ [0, l], so that α ′ (t) ≤ |∇ Σ f (γ(t))|. Since ∇f has unit length we have: 1 = |∇ Σ f | 2 γ(t) + c 2 α(t) 2 and therefore

|α ′ (t)| 2 ≤ 1 -c 2 α(t) 2 .
Fix ε > 0 and let A = {t ∈ [0, l] : α ′ (t) > 0}. Then:

l ≥ A dt ≥ A α ′ (t)dt 1 -c 2 α(t) 2 + ε ≥ l 0 α ′ (t)dt 1 -c 2 α(t) 2 + ε .
Changing variables and observing that α(0) = -1 c and α(l) = 1 c we have

l ≥ 1 c 1 -1 dx √ 1 -x 2 + ε .
Letting ε → 0 + gives l ≥ π c , as asserted. Finally, we remark that the conclusion of the Theorem holds also if the assumption on the non-negativity of the sectional curvature is replaced by the following assumptions: the Ricci curvature of Ω is non-negative, and the mean curvature of Σ is constant. We omit the details.

D

  the relative Dirichlet-to-Neumann operator. It defines another elliptic pseudodifferential operator of order one acting on Λ p (Σ), which is self-adjoint and nonnegative. These properties can easily be derived from Theorem 11 and the fact that T [p] D is related to the absolute Dirichlet-to-Neumann operator by the identity T [p] D = (-1) p(n-p) ⋆ Σ T [n-p] ⋆ Σ , where ⋆ Σ denotes the Hodge-star operator acting on forms on Σ. Denoting by ν D 1,p (Ω) the first eigenvalue of T [p] D , we have

Lemma 13 .

 13 restriction to the boundary is a Dirichlet-to-Neumann eigenfunction associated to the eigenvalue c. Let p ≥ 1. If ω ∈ F p (c) and V is a parallel vector field on R n+1 , then i V ω ∈ F p-1 (c).

( a )

 a If the equality holds, then Σ has constant positive mean curvature H = ν 1,n-1 (Ω)/n, and the restriction of f to Σ is an eigenfunction of ∆ Σ associated to the eigenvalueλ . = ν 2,0 (Ω)ν 1,n-1 (Ω). (b) If n = dim(Σ) ≥ 3 and Ω ⊂ R n+1, then the equality holds if and only if Ω is a ball.Proof. The inequality follows immediately from (c) of Corollary 15 applied to df (which has constant norm by our assumptions). We can assume that f integrates to zero on Σ. (a) If the equality holds, then f has to be a Dirichlet-to-Neumann eigenfunction associated to ν 2,0 (Ω): ∂f ∂N = -ν 2,0 (Ω)f, and (see the proof of Proposition 14) the canonical co-primitive α of df , solution of the problem δα = df, dα = 0 J ⋆ α = 0 on Σ must be a dual eigenform associated to ν D 1,1

D

  , with first eigenvalue ν D 1,0 (Ω) = ν 1,n (Ω). So, we need to show that ν D 1,0 (Ω) ≤ Vol(Σ)/Vol(Ω). Consider the 1-form α = dE. Then J ⋆ α = 0 and we can use α as a test-form for ν D 1,0 (Ω). Since i N α = ∂E/∂N, by the variational characterization (9) we get

D

  (c) = J ⋆ (δα) = 1 because δα = ∆E = 1. This shows that 1/c is an eigenvalue of T [0]D as asserted, and the associated eigenfunction is constant.

4. 4

 4 Hodge-Laplace eigenvalues: proof of Theorem 8

Theorem 19 .

 19 Let (Ω n+1 , g) a compact, connected Riemannian manifold with smooth boundary Σ. Assume that there exist a non-trivial function f ∈ C ∞ (Ω) and a number c > 0 such that:    ∇df = 0 on Ω ∂f ∂N = -cf on Σ.

  2 and Ω is a Euclidean domain. Precisely: Theorem 2. Assume that Ω has non-negative Ricci curvature and mean-convex boundary.

Then ν 1,n (Ω) ≥ (n + 1)H, where H is a lower bound of the mean curvature. If n ≥ 2, equality holds if and only if Ω is a Euclidean ball. Theorem 3. If p > n+1 2 and Ω is a Euclidean domain, then we have equality in (2) if and only if Ω is a ball.

Proof. Let φ be an eigenfunction associated to λ 1 (Σ), φ its harmonic extension to Ω and ω = d φ. Then ω is an harmonic field of degree 1. We apply the Reilly formula to ω; as ∇ω 2 ≥ 0 and σ n (Σ) = nH, we obtain:

Note that, if the equality holds, then ω must be parallel. Our curvature assumptions imply in particular that H 1 R (Ω) = 0. Therefore we can apply Proposition 14 and obtain

The lower bound (27) follows. The estimate is sharp because, for the Euclidean unit ball, we have

Now assume that ( 27) is an equality. Then ω = d φ is parallel, and we can apply Proposition 17 to f = φ. However, as we must have equalities in (28), we conclude that

and we are in the equality case of Proposition 17. So the mean curvature is constant: nH = ν 1,n-1 (Ω) and λ 1 (Σ) = ν 2,0 (Ω)ν n-1 (Ω). By assumption

and we easily obtain ν 2,0 (Ω) = σ 1 (Σ). Now, at each point of Σ, the mean curvature is always no less than the lowest principal curvature, which implies that H ≥ σ 1 (Σ) = ν 2,0 (Ω). We arrive at the inequality

By the result of Ros already cited ( [START_REF] Ros | Compact Hypersurfaces with constant higher order mean curvatures[END_REF]) we know that Vol(Σ)/Vol(Ω) ≥ (n + 1)H with equality if and if Ω is a Euclidean ball. Then Ω must be a Euclidean ball, and the proof is complete.

Biharmonic operator: proof of Theorem 10

We now consider the fourth order Steklov problem [START_REF] Escobar | An isoperimetric inequality and the first Steklov eigenvalue[END_REF] and its first eigenvalue µ 1 (Ω). As