Nina Daneu
email: nina.daneu@ijs.si

Tina Radošević

Slavko Bernik

Darko Hanžel

Matjaž Mazaj

Marjeta Maček Kržmanc

Dejan Verhovšek

Andraž Kocjan

Mirijam Vrabec

Matjaž Spreitzer

Tina Radoševič

Emmanuel Guilmeau

Thermoelectric properties of pseudobrookite-based ceramics prepared from natural Fe-Ti-rich heavy mineral sand concentrate

Keywords: thermoelectrics, Fe-Ti-oxides, natural raw materials, functional oxides, ecologic

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Waste heat energy is unavoidable in many industrial processes and could be at least partially converted into usable electricity by the installation of thermoelectric (TE) generators [START_REF] Araiz | Prospects of waste-heat recovery from a real industry using thermoelectric generators: Economic and power output analysis[END_REF]. Currently, the most efficient TE materials are based on lead, germanium, selenium, tellurium, and other toxic, scarce, and costly elements [Wei et al., 2020]. The development of new materials with high efficiency, and stability at high temperatures, produced from abundant, economic, ecological raw materials and, at reasonable costs, would boost the production and applications of thermoelectrics, and contribute to the reduction of our carbon footprint.

Metal oxide based ceramics are a promising group of thermoelectric materials because they fulfill many criteria for high-temperature applications [START_REF] Lin | Oxide thermoelectric materials: From basic Principles to Applications[END_REF]. Among the most studied oxide thermoelectrics are layered Na-and Ca-cobaltates [START_REF] Terasaki | Large Thermoelectric Power in NaCo2O4 Single Crystals[END_REF][START_REF] Shikano | Electrical and thermal properties of single-crystalline (Ca2CoO3)0.7CoO2 with a Ca3Co4O9 structure[END_REF][START_REF] Masset | Misfit-layered cobaltite with an anisotropic giant magnetoresistance: Ca3Co4O9[END_REF], Sr-titanates including SrTiO3 and layered Ruddlesden-Popper type Sr2TiO4 [Ohta et al., 2007;[START_REF] Lee | Ruddlesden-Popper phases as thermoelectric oxides: Nb-doped SrO(SrTiO3)n (n=1,2)[END_REF], doped and/or nanostructured ZnO-based compositions [START_REF] Ohtaki | High-temperature thermoelectric properties of (Zn1-xAlx)O[END_REF]Giovanelli et al., 2018;[START_REF] Guilmeau | Inversion Boundaries and Phonon Scattering in Ga:ZnO Thermoelectric Compounds[END_REF][START_REF] Labégorre | Phonon Scattering and Electron Doping by 2D Structural Defects in In/ZnO[END_REF][START_REF] Guilmeau | Inversion Boundaries and Phonon Scattering in Ga:ZnO Thermoelectric Compounds[END_REF] and few others. The main advantages of oxide ceramics in comparison to other 3 thermoelectric compounds with superior efficiency are their lower toxicity, chemical stability at higher temperatures and in oxidative atmosphere, and the possibility to prepare large amounts of materials at reasonable costs. The performance of metal oxides, i.e. their figure of merit ZT (ZT=

! ! # k 𝑇, where S is the Seebeck coefficient [μV K -1 ], s is the electrical conductivity [mΩ cm - 1 ], k is the thermal conductivity [W m -1 K -1 ], T -absolute temperature [K]
) is still fairly low. In general, oxide ceramics are characterized by average thermal conductivity but can exhibit high Seebeck coefficient, which is beneficial for application in thermoelectrics. On the other hand, oxides have low electrical conductivity which is mainly related to their low carrier concentration and less covalent bonding compared to for example state-of-the-art TE chalcogenides. Strategies for increasing the electrical conductivity and carrier concentration include doping and special microstructural engineering approaches.

Recently, compounds of the ferropseudobrookite (Fpb), FeTi2O5 -pseudobrookite (Psb), Fe2TiO5

or Fpb-Psb solid solution series, were reported as promising thermoelectrics due to their low thermal conductivity and high Seebeck coefficient [START_REF] Chen | Thermoelectric properties of Fe2(Ti1-xNbx)O5 pseudobrookite ceramics with low thermal conductivity[END_REF][START_REF] Chen | Thermoelectric properties of Fe2-xTi1+xO5 solid solutions: Influence of microcracking and Nb substitution[END_REF][START_REF] Chen | Effects of impurities on the stability of the low thermal conductivity in Fe2TiO5 ceramics[END_REF]. Theoretical calculations [START_REF] Pakizeh | Structural, electronic, magnetic and thermoelectric properties of pseudobrookite-type Fe2-xTi1+xO5 (x = 0, 0.5 and 1) compounds: DFT + U approaches[END_REF] indicated that Fe2-xTi1+xO5 (x = 0, 0.5, 1.0) phases could reach figure of merit, ZT around unity. Pseudobrookites crystallize in an orthorhombic crystal system (Cmcm space group) with neither random nor perfectly ordered Fe 2+ , Fe 3+ and Ti 4+ cations at 4c and 8f sites, as shown in Figure 1a. In ferropseudobrookite (FeTi2O5, Fpb), most of the Fe ions are in the ferrous (divalent, Fe 2+ ) state and occupy the 4c site.

The 8f sites are progressively occupied with higher amounts of Fe as the compositions approach pseudobrookite (Fe2TiO5, Psb) [START_REF] Guo | Crystal structure and cation distributions in the FeTi2O5-Fe2TiO5 solid solution series[END_REF]. Fpb starts to form already above 585°C and is stable approximately up to 1150°C. The Fpb-Psb series is one of the three solid solution series that occur in the FeO-Fe2O3-TiO2 phase diagram [Lindsley, 1991]; the other two are magnetite -4 ulvöspinel (Fe3O4-Fe2TiO4) series with the spinel-type structure and ilmenite-hematite (FeTiO3-Fe2O3) series with the corundum-type structure (Figure 1b). Pseudobrookites for thermoelectric applications are usually synthesized from high-purity starting oxides (FeO,Fe2O3,TiO2) by a combination of solid-state reaction and conventional pressureless solid-state sintering [START_REF] Chen | Thermoelectric properties of Fe2(Ti1-xNbx)O5 pseudobrookite ceramics with low thermal conductivity[END_REF][START_REF] Chen | Thermoelectric properties of Fe2-xTi1+xO5 solid solutions: Influence of microcracking and Nb substitution[END_REF][START_REF] Chen | Effects of impurities on the stability of the low thermal conductivity in Fe2TiO5 ceramics[END_REF]. Natural Fe-Ti-rich powders can represent an alternative source for the preparation of pseudobrookite ceramics, especially because they are abundant and inexpensive raw materials. These powders are obtained from Fe-Ti-rich heavy mineral sands (HMS), naturally occurring concentrates of minerals with higher density. HMSs are most commonly used as a source of titanium (Ti) or titania (TiO2) for applications in aerospace, defense, medical, pharmaceutical, food, paint, adhesive, and other industries [START_REF] Jones | Mineral Sands: an Overview of the Industry[END_REF]. The HMS accumulations form by water and wind-driven erosion processes and the characteristic constituent minerals include ilmenite (FeTiO3; ρ = 4.3-4.6 g/cm 3 depending on the alteration rate), zircon (ZrSiO4; ρ = 4.6-4.7 g/cm 3 ), rutile (TiO2; ρ = 4.23 g/cm 3 ), sillimanite (Al2SiO5; ρ = 3.24 g/cm 3 ), monazite (Ce, La, Nd, Th)PO4, ρ = 4.8-5.5 g/cm 3 ), diamond (C, ρ = 3.5 g/cm 3 ), cassiterite (SnO2; ρ = 6.9 g/cm 3 ), garnet (X3Y2(SiO4)3 with X = Mg, Fe, Mn, Ca and Y = Al, Fe, Cr; ρ = 3.5-4.3 g/cm 3 ) and others, depending on the local geological environment. The HMS is separated into different mineral phases by a combination of magnetic, electrostatic, and gravity techniques [Rejith and Mayappan, 2018]. The Fe-bearing magnetic minerals are recovered by a combination of magnetic and electrostatic separation and this fraction contains mainly ilmenite (FeTiO3) possibly also some rutile and hematite. In addition, the magnetic fraction contains alteration products of ilmenite, in which Fe is partially oxidized from Fe 2+ to Fe 3+ . Iron may also be, to some extent, leached out of the grains to form so-called leached ilmenite (Fe2+xTi3O9), pseudorutile (Fe2Ti3O9; Fe2O3×3TiO2), and leucoxene ((Ti,Fe)3(O)6(OH)6)) [START_REF] Mücke | Giant thermoelectric Seebeck coefficient of a twodimensional electron gas in SrTiO3[END_REF]. Ilmenite grains occasionally contain exsolved rutile needles [START_REF] Ramdohr | The ore minerals and their intergrowths, 3 rd Edition[END_REF], which are indicative of oxidation processes during the crystallization of the primary rock [START_REF] Rečnik | Topotaxial reactions during the genesis of oriented rutile/hematite intergrowths from Mwinilunga (Zambia)[END_REF][START_REF] Stanković | Topotaxial reactions during oxidation of ilmenite single crystal[END_REF].

In this work, we explored the possibility of preparing pseudobrookite-based ceramics for thermoelectric applications from natural Fe-Ti-rich concentrate, which is otherwise used for the extraction of TiO2. Even though the synthesis of advanced ceramics from high-purity starting chemicals is typically recommended to avoid the effect of impurities, we show that natural Fe-Tirich HMS concentrate with ~95% purity is a promising raw material for the production of pseudobrookite ceramics with low thermal conductivity.

Experimental

Preparation of the starting powders

For our experiments, we used natural Fe-Ti-rich HMS concentrate provided by Cinkarna Metallurgical and Chemical Industry Celje, INC. The starting powder without additional treatment is the so-called as-received powder. The sand for the preparation of the concentrate originated from heavy mineral placers in Mozambique and was preliminarily milled in the Cinkarna Celje company to reach 95% of particles with a size below 40 μm for further use as raw material for the production of titania, TiO2. The as-received powder contained around 95.4% of Fe-Ti-rich phases with 53.0 wt% of TiO2, 23.1 wt% of Fe2O3, and 19.3 wt% of FeO. According to the weight fractions, the Fe:Ti atomic ratio in the as-received material was 54.4:45.6 (1.19). Detailed chemical analysis of the as-received powder is given in the Supporting Materials (Table S1). Three additional starting powders were prepared by pre-oxidation of the as-received powder at 600°C, 700°C and 800°C for 1 hour in an air atmosphere. Pre-oxidation of ilmenite is usually applied for increased and more efficient extraction of TiO2 from the raw material [START_REF] R12685 | Effect of preoxidation and sintering on properties of ilmenite concentrates[END_REF], in this work we used the procedure for the modification of the ferrous/ferric ratio in the starting powders. The preoxidation temperatures were selected based on the results of thermogravimetric analysis of the asreceived powder (S2). During the pre-oxidation procedure, the powders were placed in a larger alumina crucible and heated to the targeted temperature in a chamber furnace in an air atmosphere with a heating/cooling rate of 10°C/min.

Synthesis of the compacts for thermoelectric applications

The starting powders (the as-received powder and the pre-oxidized powders) were sintered into dense compacts using two different approaches. For conventional sintering (CS), the powders were uniaxially pressed into cylindrical pellets under 150 MPa. The pressed compacts were placed in an alumina crucible and sintered in a tube furnace in a protective atmosphere (Ar flow) at 1200°C for 5h with a heating/cooling rate of 10°C/min. Cylindrical pellets with two dimensions (diameter x height; ϕ x h) were prepared; the 8 mm x 3 mm pellets were used for structural and microstructural analyses, and measurements of absolute thermoelectric parameters, whereas the 10 mm x 10 mm pellets were used for measurements of actual thermoelectric response over a wide range of temperature gradients. For the second sintering procedure, we used an SPS furnace (FAST, Dr.

Sinter FAST515-S, Sumimoto FAST Syntex Ltd., Japan). The powders were placed in a cylindrically shaped graphite die with a 10 mm diameter and pressed with 50 MPa. SPS sintering was performed in Ar atmosphere under 150 MPa pressure and with a heating rate of 100°C/min to the final temperature of 1100°C, dwell time was 5 minutes. With SPS we also prepared compacts with two dimensions: smaller ones were prepared from 0.5 g of the powders and were used for structural and microstructural characterization, whereas the larger ones were prepared from 1.5 g of the powders and were used for measurements of thermoelectric properties.

Characterization methods

Phase composition of the samples was analyzed by X-ray powder diffraction (XRD; Bruker AXS D4 Endeavor, Germany). The diffraction patterns were recorded in the angular range 10-80° 2Θ with 0.04° step size and 3s/step. The data were analyzed by X'Pert High Score Plus diffraction software. The valence state and local environment of iron in different phases in the samples were analyzed using Mössbauer spectroscopy (MS; Wissel Mössbauer spectrometer). For each measurement, we prepared around 70 mg of powder (starting powders and crushed sintered pellets) mixed with BN (boron nitride) powder. The Mössbauer spectra were recorded at room temperature using a constant acceleration Wissel spectrometer, in transmission mode with Co-57 source in Rh 8 matrix. Velocity calibration was performed using a thin α-Fe foil. The isomer shifts are expressed relative to α-Fe at room temperature. The spectra were fitted with the analysis code RECOIL [START_REF] Lagarec | Recoil -Mössbauer spectral analysis software for Windows[END_REF]] using Lorentzian line shape and the following parameters were refined: isomer shift (δ), quadrupole splitting (Δ), quadrupole shift (ε), magnetic hyperfine field (Bhf), signal line width (Γ) and relative area of the peaks (S). Microstructural analyses of the polished cross-sections of the samples were performed using scanning electron microscopy (SEM; JSM 7600F, JEOL, Japan) operated at 15 kV and equipped with energy dispersive X-ray spectrometer (EDXS; INCA Oxford 350, Oxford Instruments, UK) for the determination of chemical composition (interaction volume in the range of μm 3 ). Absolute density (ρabs) of the sintered compacts was measured using Densitec (Metar Sa, Rossens, Switzerland). The measurement is based on Archimedes principle, where the mass of the sample is measured in air and in oil. Relative density (ρrel) of the samples was calculated relative to the theoretical density of pseudobrookite (ρPsb= 4.4416 g/cm 3 ). Thermoelectric properties of the sintered compacts were determined by two methods. Absolute values of the electrical resistivity (ρ) and Seebeck coefficient (S) were measured simultaneously in the temperature range from 300 K up to 800 K using an ULVAC-ZEM3 instrument under partial helium pressure. The measurements were performed from 2 × 3 × 10 mm 3 pieces cut from the sintered pellets. The thermal conductivity (κ) was determined as the product of the geometrical density, the thermal diffusivity (Netzsch LFA457), and the theoretical heat capacity using the Dulong-Petit approximation. The lattice contribution to the thermal conductivity (κL) was determined by subtracting the estimated electronic component (κe) from the measured total thermal conductivity, κ. The electronic contribution, κe, was derived from the Wiedemann-Franz law, 𝜅 $ = 𝐿𝜎𝑇, where the Lorenz number, L, was approximated from the Seebeck coefficient using the simplified expression L = 1.5 + exp(-|S|/116). Note that ρ, S and κ were measured in the direction perpendicular to the SPS pressure direction. Thermoelectric parameters of the sintered pellets, i.e. the Seebeck coefficient (S), electrical conductivity (σ), and total thermal conductivity (κ) were also simultaneously measured in dependence on the temperature gradient in the temperature range from 40 °C to 450-500 °C in vacuum (10 -2 -10 -3 Pa) using a custom-built Z-meter based on the "large-delta-T" principle. The results of these measurements are values in a given temperature gradient between the hot and the cold side of the sample, which thus reflect the actual thermoelectric response of the material in conditions of application.

Results and discussion

Characterization of the starting powders

The Fe-containing phases in the starting powders (the as-received powder and the pre-oxidized powders) were analyzed by Mössbauer spectroscopy (MS). The results are shown in Figure 2 and the refined Mössbauer parameters are given in Table S2. The as-received powder is altered ilmenite [START_REF] Daneu | Pseudobrookite-based ceramics with low thermal conductivity prepared from Fe-Ti-rich heavy mineral sand concentrate 28[END_REF] which contains only about half of iron in the ferrous state. Most of the Fe 2+ is in the form of ilmenite (FeTiO3, Ilm) with octahedrally coordinated Fe atoms, a minor amount of Fe 2+ is in ulvöspinel (Fe2TiO4; Usp), where Fe atoms are tetrahedrally and octahedrally coordinated [Ono et al., 1968]. The remaining Fe is in the trivalent form (Fe 3+ ), mainly as pseudorutile (Fe2Ti3O9; Prt), which is a common alteration product of ilmenite [START_REF] Mücke | Giant thermoelectric Seebeck coefficient of a twodimensional electron gas in SrTiO3[END_REF]. Some Fe 3+ is also bound to hematite (Fe2O3; Hem). After pre-oxidation of the as-received powder at 600 °C for 1 hour, the powder contains significantly less ferrous iron, only as ilmenite. The amounts of Fe 3+ phases (Prt and Hem) increase after oxidation at 600°C and after pre-oxidation at 700°C and 800°C, the samples contain only Fe 3+

as Prt and Hem. The result indicates that oxidation of iron is completed after 1 hour at 700°C.

11

XRD was used to detect the non-ferrous phases in the samples (Figure 3). The XRD pattern of the as-received powder is characterized by fairly sharp peaks, which belong to ilmenite, pseudorutile, and hematite. The presence of other phases (e.g. ulvöspinel, which was detected by MS) was below the detection limit of XRD. After pre-oxidation treatments between 600 and 800°C, the reflections of ilmenite decrease, while those of pseudorutile increase and become wider indicating increasing crystallization of pseudorutile (Fe2Ti3O9; 3TiO2•Fe2O3) composed of finely intergrown Fe2O3 and TiO2. In addition, increasing peaks from hematite and rutile that form by progressive oxidation of ilmenite are also detected: The results of MS and XRD analyses of the starting powders suggest that ilmenite grains in the asreceived powder start to oxidize already below 600°C and partially transform to rutile and hematite and partially to pseudorutile [START_REF] Xiao | Phase transitions, micro-morphology and its oxidation mechanism in oxidation of ilmenite (FeTiO3) powder[END_REF][START_REF] Cheng | Microstructure and chemical transformation of natural ilmenite during isothermal roasting process in air atmosphere[END_REF] via following reactions:

2 FeTiO3 + ½ O2 → Fe2O3 + 2TiO2
(1) substitute in the Fe 2+ sites, whereas the trivalent cations (e.g. Al) most likely occupy the Ti 4+ sites, which requires charge compensation through the formation of Fe interstitials of O vacancies [START_REF] Kuganathan | Theoretical Modeling of Defects, Dopants, and Diffusion in the Mineral Ilmenite[END_REF]. SEM analyses also revealed the presence of accessory minerals like zircon, monazite, quartz and alumosilicates; however, these particles were small and fairly rare.

Some examples are shown in Figure S3.

After pre-oxidation at different temperatures, the size and shape of the particles remain similar (Fig. 4b-d), however, most of the grains contain oriented rutile exsolutions as a result of topotaxial recrystallization during oxidation of ilmenite [START_REF] Rečnik | Topotaxial reactions during the genesis of oriented rutile/hematite intergrowths from Mwinilunga (Zambia)[END_REF][START_REF] Stanković | Topotaxial reactions during oxidation of ilmenite single crystal[END_REF]. In addition, these grains are characterized by a thin layer of hematite on the surface as a result of Fe 3+

out-diffusion during the process. The pre-oxidized powders contain grains with different density and size of the rutile lamellae. The transformation depends on the size and crystallinity of the starting grains; the larger and more compact grains oxidize slower [START_REF] Rečnik | Topotaxial reactions during the genesis of oriented rutile/hematite intergrowths from Mwinilunga (Zambia)[END_REF][START_REF] Stanković | Topotaxial reactions during oxidation of ilmenite single crystal[END_REF] in comparison to smaller grains [START_REF] Fu | Phase Transitions and Reaction Mechanism of Ilmenite Oxidation[END_REF] and hence, the larger ilmenite particles transform to grains with larger rutile lamellae (Fig. 4e,left). These lamellae are well resolved by SEM and are also detected by XRD as rutile phase. On the other hand, altered grains with higher degree of porosity transform to finer (nanoscale) oriented rutile exsolutions (Fig. 4e, right), which are detected as pseudorutile by XRD.

14

The as-received powder and the pre-oxidized powders with different Fe 2+ : Fe 3+ ratios were used as the starting powders for the synthesis of dense Fe-Ti-oxide compacts using conventional sintering (1200°C/5h) and spark plasma sintering (1100°C/5min) in protective Ar atmosphere.

Structural and microstructural characterization of the sintered compacts

Mössbauer spectra of the compacts prepared from different starting powders after conventional sintering and SPS are shown in Figure 5 and the calculated Mössbauer parameters are given in Table 2. The sintered compacts prepared from as-received powder have similar compositions after both sintering procedures (CS and SPS). The only Fe 2+ -containing product is ferropseudobrookite [START_REF] Guo | Crystal structure and cation distributions in the FeTi2O5-Fe2TiO5 solid solution series[END_REF], whereas the presence of some ilmenite indicates that it remained partially unreacted during the sintering in Ar. Both ferric oxides from the starting powder, pseudorutile and hematite, completely transformed into pseudobrookite. The presence of ferropseudobrookite is observed also in the samples prepared from the powder pre-oxidized at 600°C, which contained some residual ilmenite. Interestingly, conventionally sintered compacts prepared from the powders pre-oxidized at 700 and 800°C contain some ferropseudobrookite even though the powders were fully oxidized in the pre-oxidation step (Table 1). This suggests that some Fe 3+ from Fe2TiO5 was reduced back to Fe 2+ during the 5 h sintering at 1200°C in argon.

The reduction of iron did not occur in the samples sintered by SPS due to the lower sintering temperature (1100°C) and significantly shorter sintering time of only 5 minutes. The only Fe-phase in these samples is pseudobrookite. The samples prepared from the as-received powder via both sintering procedures (CS and SPS) contain only phases on the Psb-Fpb tie-line and some ilmenite. No rutile was detected in these samples. On the other hand, all samples prepared from pre-oxidized powders contain rutile that 18 formed during the pre-oxidation stage. The intensity of rutile reflections in the CS samples is similar in all samples prepared from powders pre-oxidized at different temperatures, whereas the intensity of rutile reflections in compacts prepared by SPS increases with pre-oxidation temperature (see the detail on the right side of Figure 6). This suggests that, during conventional sintering, some rutile reacted with Psb, triggering a reduction of Fe 3+ to Fe 2+ and the formation of Fpb according to the following reaction:

Fe2TiO5 + 3TiO2 → 2FeTi2O5 + ½ O2 (3)
The results of MS and XRD analyses have shown that the samples prepared by conventional sintering and SPS from the same starting powders have comparable phase composition; however, microstructural analyses (Figure 7) reveal that the samples prepared by those different processing routes have different grain sizes, densities (Table 3), and distribution of phases. The microstructures of the samples after conventional sintering are shown in stable Fe 2+ :Fe 3+ ratios. Some grains with an even higher amount of Fe were found, whereas grains with a Fe:Ti ratio of 1 (as expected for ilmenite) were not found even though the presence of ilmenite was indicated by MS and XRD. Microstructural characteristics of the sample suggest that the starting powder, composed of up to 40 μm large particles of pseudorutile, ilmenite, and some ulvöspinel and hematite locally recrystallized to grains with Psb-Fpb composition, and the processes are governed by slow diffusion. Samples prepared from pre-oxidized powders have higher porosity, relative density between 83 and 86%, and grain size below 10 microns. In these samples, iron is prevalently in the Fe 3+ form and hence the matrix phase grains have a more uniform Fe:Ti ratio, which is most likely a solid solution between pseudobrookite and ferropseudobrookite. Typical for these samples is the presence of intergranular rutile grains, which formed by aggregation of rutile lamellae that formed in the ilmenite grains during the pre-oxidation. During high-temperature sintering, the rutile lamellae inhibited the diffusion processes and therefore, the matrix (Psb-Fpb) grains remained smaller in comparison to the sample prepared from as-received powder without the rutile lamellae.

Samples sintered by SPS (Figure 7b) have smaller grain sizes and higher relative densities, above 95%, in comparison to conventionally sintered samples. The microstructural characteristics are a consequence of the fast heating rate under the applied pressure. Similar to the CS samples, the SPS sample prepared from the as-received powder has the largest grain size, and also here, areas with two different gray levels (Z-contrast in back-scattered images) are observed. The majority of the sample has Fe:Ti ratio of around 0.8 (darker regions, mostly Fpb), whereas the brighter regions have Fe:Ti ratio of around 1.5 (brighter regions; mostly Psb). This confirms that two solid solutions with stable composition are formed during the sintering by local recrystallization.

The samples prepared from pre-oxidized powders are composed of pseudobrookite matrix with micron and sub-micron rutile inclusions that occur as intra-and inter-granular particles. Regions with different concentrations of rutile needles are observed depending on the composition of the particle in the starting powder, e.g. pseudorutile or ilmenite with rutile needles, confirming that densification of the sample includes only local recrystallization.

Measurements of thermoelectric properties

The electrical and thermal transport properties of the three series of samples, measured between values of the Seebeck coefficient of respectively 27 and 52 µV K -1 at room temperature (Figure 8a). It increases linearly with temperature up to around 85-90 µV K -1 at 800 K. Such a low magnitude of | 𝑆 | and its linear increase with temperature is in agreement with the mixed valence Fe 2+ /Fe 3+ observed in those two samples (Table 2). Such behavior is typically observed in electrondoped TiO2, SrTiO3, and CaMnO3, where mixed valence Ti 3+ /Ti 4+ or Mn 3+ /Mn 4+ promotes electron conduction [X. [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF][START_REF] Wang | Record high thermoelectric performance in bulk SrTiO3 via nano-scale modulation doping[END_REF][START_REF] Lu | High-Figure-of-Merit Thermoelectric La-Doped A-Site-Deficient SrTiO3 Ceramics[END_REF]Srivastava et al., 2018;[START_REF] Daneu | Pseudobrookite-based ceramics with low thermal conductivity prepared from Fe-Ti-rich heavy mineral sand concentrate 29[END_REF]. Surprisingly, the electrical resistivity (Figure 8b) decreases with temperature for both samples, which suggests the carrier concentration remains relatively low and/or microstructural features, such as the existence of many interfaces, also govern the electrical conduction.

The samples, sintered by SPS from powders pre-oxidized at 600°C, 700°C or 800°C, have larger absolute values of the Seebeck coefficient, which increase with the annealing temperature from 250 µV K -1 for the 600°C-SPS sample to 400 µV K -1 for the 700°C-SPS sample at 300 K and 616 µV K -1 at 500 K for the 800°C-SPS sample. This evolution can be directly linked to the decrease of Fe 2+ content from 52% in the raw sample to 2% in the 600°C-SPS sample and 0% in the 700°C-SPS and 800°C-SPS samples (Table 2b). The largest amount of rutile phase in those two samples can also affect the Seebeck coefficient. Note that the electrical resistivity in this series increases with the annealing temperature, in agreement with the increase in the absolute value of | 𝑆 | .

The samples, sintered conventionally at 1200°C, do not exhibit significant and regular variations of the Seebeck coefficient with the annealing temperature. All | 𝑆 | values range between 325 and 375 µV K -1 at 450 K. This trend is correlated with the existence of Fe 2+ /Fe 3+ mixed valence in those samples (Table 2), due to the reduction induced by the high-temperature sintering. The magnitude of the electrical resistivity does not vary significantly with the annealing temperature which is in good agreement with the Seebeck coefficient values. Note that the largest porosity in this series of samples does not significantly influence the magnitude of the electrical resistivity compared with the two other series of samples. The thermal conductivities measured by the Z-meter are shown in supporting materials (S5b). As they were determined in a large DT regime, they are lower as compared to the values determined by the use of the laser flash method, ranging from about 2.2W/mK to about 0.7W/mK. It is also worth mentioning that thermal conductivity is rather constant in all the samples regardless of the temperature gradient across the sample. The fact that quite high Seebeck coefficient and quite low thermal conductivity of the samples are constant in a relatively broad range of temperature gradients from about 30K to even about 400K (i.e. temperatures of the hot side from 45°C to 450°C, respectively), is encouraging for the application of these materials. The differences between the results obtained by ZEM-3 and the Z-meter can be explained by the complexity of the samples due to their inhomogeneous composition and microstructure, which is likely more expressed in the case of larger samples measured by the Z-meter under large DT than in the case of much smaller samples measured by ZEM-3 at small DT and thus affects the obtained results.

Conclusions

Figure 1 :

 1 Figure 1: (a) Crystal structures of ferropseudobrookite (Fe 2+ Ti2O5) and pseudobrookite (Fe2 3+ TiO5). The Fe 2+ , Fe 3+ and Ti 4+ may reside in any of the available octahedral positions (8f and 4c), therefore the end-phases form a complete solid solution. (b) FeO-TiO2-Fe2O3 phase diagram with the main phases and three solid solution series, which are stable at different temperatures.

Figure 2 :

 2 Figure 2: Mössbauer spectra of the starting powders, the as-received powder and the powders after preoxidation at different temperatures for 1h in air atmosphere.

Figure 3 :

 3 Figure 3: XRD patterns of the as-received powder and powders pre-oxidized at different temperatures.

  sections of the as-received and pre-oxidized powders are shown in Figure 4.

Figure 4 :

 4 Figure 4: (a) As-received powder contains sharp-edged grains with variable sizes, and different Fe:Ti ratios as revealed by EDS mapping. (b-d) Particles after pre-oxidation at different temperatures. Ilmenite grains with different degree of crystallinity gradually recrystallize to Rt-Hem intergrowths with (e) coarser or finer (f) crystallites, the latter are characteristic for Prt.

Figure 5 :

 5 Figure 5: Mössbauer spectra of the different starting powders (as-received and pre-oxidized) after (a) conventional sintering and (b) SPS.

Figure 6 :

 6 Figure 6: XRD patterns of the starting powders after (a) conventional sintering and (b) SPS.

Figure 7a .

 7a The sample prepared from as-received powder has the highest density (89.6 %) and the largest grain size (~ 10 -50 microns) among the conventionally sintered samples. Areas with different gray levels in backscattered images indicate areas with different Fe:Ti ratios. According to MS and XRD, this sample contains mainly pseudobrookite, ferropseudobrookite, and some ilmenite. These phases have theoretical Fe:Ti ratios of 2, 0.5, and 1, respectively. SEM/EDS analyses have shown that the darker regions, which are prevailing in the sample, have a fairly constant Fe:Ti ratio of around 0.8 (Fpb:Psb ~ 60:40), whereas the brighter areas contain more Fe and have the Fe:Ti ratio of around 1.5 (Fpb:Psb ~ 10:90). This indicates that Psb and Fpb form two solid solutions with

Figure 7 :

 7 Figure 7: Microstructures of the as-received powder and pre-oxidized samples after (a) conventional sintering (1200°C/5h) and (b) SPS (1100°C/5min). Higher magnification micrographs are presented in the second row.

Figure 8 :

 8 Figure 8: Thermoelectric properties of untreated and pre-oxidized powders after conventional sintering and SPS. (a) Seebeck coefficient, (b) electrical resistivity, (c) thermal conductivity, and (d) power factor, as a function of temperature.

Table 3 :

 3 Absolute and relative densities of samples prepared from different starting powders after

			(a) conventional sintering and (b) SPS.
		(a) Conventional sintering		(b) SPS
		Measured density	Relative density	Measured density	Relative density
	Sample			Sample
		[g/cm 3 ]	[%]		[g/cm 3 ]	[%]
	as-received	4.071	89.6	as-received	4.238	95.1
	powder			powder
	600°C/1h	3.648	83.0	600°C/1h	4.244	96.8
	700°C/1h	3.717	84.0	700°C/1h	4.263	96.8
	800°C/1h	3.796	86.2	800°C/1h	4.274	97.1

Acknowledgements

The authors gratefully acknowledge financial support by the Slovenian Research Agency under the Project No. J1-9177, Programme P2-0091, and Programme P1-0112. EG and SB thanks Campus France for financial support (PROTEUS 44203UB).

In this work, we used natural Fe-Ti-rich heavy mineral sand as a raw material for the preparation of ceramics for thermoelectric applications. The phase composition and microstructural properties of the sintered compacts and the related thermoelectric properties depend on the starting Fe:Ti ratio, the application of the pre-oxidation procedure of the as-received powder, and the sintering process (conventional sintering or SPS). All samples exhibit relatively low thermal conductivity for oxides, but the ZT values of all samples are very low due to the low PF values. The best properties were obtained for the SPS sintered powder that was pre-oxidized at 600 °C. Further investigations are required to optimize the electrical conduction by balancing the starting Fe:Ti ratio to avoid the formation of rutile and appropriate doping strategies. The correlation between Mössbauer spectroscopy and electrical transport properties data indicates that the mixed valence Fe 2+ /Fe 3+ is the key parameter to generate optimized thermoelectric performances. Nevertheless, the measured values of thermoelectric parameters obtained on samples prepared from natural powders are comparable to those obtained on samples prepared from pure (>99%) starting chemicals [START_REF] Chen | Thermoelectric properties of Fe2(Ti1-xNbx)O5 pseudobrookite ceramics with low thermal conductivity[END_REF][START_REF] Chen | Thermoelectric properties of Fe2-xTi1+xO5 solid solutions: Influence of microcracking and Nb substitution[END_REF]. The results of this study suggest that natural Fe-Ti-rich heavy sands, which represent abundant, nontoxic, and low-cost raw materials, are potentially interesting for the preparation of ceramics for thermoelectric application after some additional improvements.


A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS

We prove a positive mass theorem for complete Kähler manifolds that are asymptotic to the complex hyperbolic space.

Introduction

The aim of this paper is to provide a positive mass theorem in the realm of asymptotically complex hyperbolic Kähler manifolds, extending previous results by M. Herzlich [Her] and Boualem-Herzlich [BH]. Before explaining them, let us recall the history of the subject.

The classical positive mass theorem finds its roots in general relativity [ADM] and deals with asymptotically Euclidean manifolds, namely complete Riemannian manifolds (M n , g) whose geometry at infinity tends to that of the flat Euclidean space : M is diffeomorphic to R n outside a compact set and g goes to g R n at infinity. Under a nonnegativity assumption on the curvature (Scal g ≥ 0), the positive mass theorem roughly asserts that such manifolds possess a global Riemannian invariant, which is called a mass, which is obtained by computing the limit of integrals over larger and larger spheres, which is a non-negative number and which vanishes only when the manifold is isometric to the model flat space. This "Euclidean mass" is given in some where the sphere, Hodge star, the divergence and the trace are defined with respect to the Euclidean metric g R n at infinity. It is not obvious that this quantity depends only on g and not on the chart but [Bart] proved it. We refer to [START_REF] Schoen | On the proof of the positive mass conjecture in general relativity[END_REF][START_REF] Schoen | Proof of the positive mass theorem[END_REF][START_REF] Witten | A new proof of the positive energy theorem[END_REF][START_REF] Bartnik | The mass of an asymptotically flat manifold[END_REF][START_REF] Lee | The Yamabe problem[END_REF] for details and "classical" proofs and to [Loh] for a more recent and more general treatment.

The mathematical interest of such a theorem is the rigidity result it involves: under a nonnegativity assumption on the curvature, it asserts a model metric at infinity (the Euclidean metric, here) cannot be approached at any rate, the obstruction being precisely the mass. Another striking feature of this theorem is its role as a key step in the first proof of the Yamabe theorem [LP] ; see also [ST] for a nice application of the positive mass theorem to obtain a rigidity result for compact manifolds with boundary. It is possible to extend these ideas behind this theorem in several contexts, involving different models at infinity. The authors of [MO, AD, CH] studied the case of manifolds whose model at infinity is the real hyperbolic space RH n . More sophisticated models are also considered in [Dai, Min]. The notion of real hyperbolic mass introduced in [CH] is at the very root of our work so we need to explain what it looks like (note it is more general than what follows). Basically, while the Euclidean mass ( 1) is a single number, the real hyperbolic mass appears naturally as a linear functional on some finite dimensional vector space N , attached to the model at infinity, RH n . More precisely, [CH] defines N as the set of functions u on RH n such that Hess RH n u = ug RH n and the mass linear functional is given by (2)

µ g (u) = - 1 4 lim R-→∞ S R * [(div g + d Tr g) u -Tr(g -g RH n )du + (g -g RH n )(grad u, .)] ,
where the right-hand side is computed with respect to the real hyperbolic metric g RH n at infinity. It turns out that N can be interpreted as the set of parallel sections for some natural vector bundle E, endowed with a natural connection ∇ RH preserving a natural Lorentz metric h : if you see RH n as a hypersurface in Minkowski space R n,1 , then (E, h) is simply the restriction of the tangent bundle T R n,1 and ∇ RH is induced by the flat connection on Minkowki space ; from this point of view, N identifies naturally with R n,1 . We will give more details on this picture in the first section of this text and explain why it is natural that the mass belongs to the dual of this space N and why the formula above is a relevant geometric invariant. The basic idea is that any Riemannian manifold carries a natural connection ∇ RH that is flat iff the manifold is (locally) hyperbolic ; the mass appears naturally as an obstruction to the construction of parallel sections for this connection.

The model at infinity we wish to consider is the complex hyperbolic space CH m , which is a counterpart of the real hyperbolic space RH n in complex geometry. Up to scale, it is indeed the unique simply-connected complete Kähler manifold with constant holomorphic sectional curvature. To be concrete, let us recall that CH m can be seen as the unit ball in C m , endowed with its standard complex structure J and with the Kähler metric

g CH m = g CH m = 1 (1 -s 2 ) 2 ds 2 + (Jds) 2 + s 2 (1 -s 2 )g F S ,
where s is the radial coordinate in C m and g F S is the Fubini-Study metric of CP m-1 , pulled-back to C m \{0}. Setting s = tanh r, we obtain CH m as the complex manifold C m endowed with

g CH m = dr 2 + (sinh 2r) 2 η 2 + (sinh r) 2 g F S
where η = -Jdr sinh 2r is the standard contact form on S 2m-1 . This is an Einstein metric with scalar curvature -4m(m + 1) -the holomorphic sectional curvature is -4. The most useful description of CH m for us is yet another one. Let C m,1 denote the vector space C m+1 endowed with a Hermitian form h of (complex) signature (m, 1). Then the level set h = -1 in C m,1 , endowed with the induced metric, is a well-known Lorentz manifold, called Anti-de-Sitter space AdS 2m,1 . The quotient of AdS 2m,1 by the scalar action of S 1 is then a Riemannian manifold and it is precisely CH m .

In this paper, we define asymptotically complex hyperbolic manifolds as complete Kähler manifolds (M 2m , g, J) such that:

(i) M minus a compact subset is biholomorphic to CH m minus a ball and, (ii) through this identification, gg CH m = O(e -ar ) with a > m + 1 2 (in C 1,α ). Note the definition in [Her], while apparently weaker, is indeed equivalent (cf. the remark after Definition 3.1). The papers [Her, BH] prove rigidity results about asymptotically complex hyperbolic manifolds that look like the rigidity part (the vanishing mass part) of a positive mass theorem. What is the mass in this setting ?

In complete analogy with the real hyperbolic case, we will see the mass as a linear functional on some natural finite-dimensional vector space N attached to the model at infinity, CH m . This vector space N is best described as a set of parallel sections for some natural connection ∇ CH on some natural vector bundle E over CH m . The vector bundle E is indeed Λ 2 J CH m ⊕ T CH m ⊕ R and the relevant connection comes from the flat connection on C m,1 . Details will be given in the text. To keep this introduction short, let us just point out that N identifies naturally to the vector space Λ 2 J C m,1 of J-invariant 2-forms on C m,1 and also admits a description as the set of functions u satisfying a natural third-order equation. Then we define the complex hyperbolic mass by

(3) µ g (u) = - 1 4 lim R-→∞ S R * (div g + d Tr g) u - 1 2 Tr(g -g CH m )du ,
where everything on the right-hand side is computed with respect to the complex hyperbolic metric g CH m at infinity. The following positive mass theorem holds in this context.

Theorem 0.1 -Let (M, g, J) be a spin asymptotically complex hyperbolic manifold with Scal g ≥ Scal g CH m . When the complex dimension of M is even, we also assume that M is contractible. Then µ g is a well defined linear functional, up to an automorphism of the model. It vanishes if and only if (M, g, J) is the complex hyperbolic space.

The mass also satisfies a nonnegativity property : it takes non-negative values on some distinguished orbits of the action of P U (m, 1). Under these assumptions, the mass may very well take infinite values : we then decide that is infinite and the finiteness of the mass does not depend on the choice of the chart at infinity. A simple criterion for the mass to be finite is that the rate of fall-off to the model metric at infinity be sufficiently fast : a ≥ 2m + 1. A consequence of the positive mass theorem is that if a > 2m + 1, then (M, g, J) is complex hyperbolic ; it is the object of [Her, BH]. Our viewpoint yields a different proof, maybe more direct.

The spin assumption is classical : we need spinors to implement Witten's techniques [Wit]. The additional topological assumption in the even dimensional case is quite technical but could be weakened ; we refer to the text for a more precise statement. This assumption was already used in [BH]. Basically, Witten's techniques require some "special" spinors to exist on the model. Complex hyperbolic spaces of odd dimension possess such distinguished spinors, called Kählerian Killing spinors. Such spinors do not exist in the even dimensional case, so we need an extra trick. This has a cost : an additional assumption.

Let us make a general remark. The previous positive mass theorems were very related to -if not completely immersed into -physical ideas. The asymptotically complex hyperbolic realm does not share this feature (yet). We feel it is all the more interesting to observe that the very physical idea of a positive mass theorem carries over to purely mathematical settings. It might indicate that there is a general mechanism, waiting for new applications.

The structure of the paper is as follows. In a first section, we will discuss the real hyperbolic case and explain how the introduction of a natural hyperbolic connection gives a slightly different proof of the hyperbolic positive mass theorem in [CH]. This case will also serve as a helpful guide for the complex case. In a second section, we will introduce the complex hyperbolic connection and describe its basic features and in particular how it interacts with the so-called Kählerian Killing spinors. A third section is devoted to the proof of the complex hyperbolic positive mass theorem in odd complex dimensions. A short fourth section will describe how to extend these arguments to the even dimensions. Finally, an appendix describes an example.

Acknowledgements. The authors would like to thank Marc Herzlich for bringing the problem to their attention, but also Olivier Biquard, Elisha Falbel and Paul Gauduchon for useful discussions.

1. The real hyperbolic case.

In this section, we briefly review some aspects of [CH]. The result mentioned here is not new, but we feel that the reformulation we propose might be useful. It is indeed quite simple and turns out to generalize to the complex hyperbolic setting. The basic question we address is the following : given a Riemannian manifold that looks like the hyperbolic space, how can one ensure that it is actually the hyperbolic space ? The approach suggested here consists in finding a connection characterizing the hyperbolic space, in that it is flat only on a (locally) hyperbolic manifold, and then try to build parallel sections for this connection.

Let us start with a Riemannian manifold (M n , g). We consider the vector bundle E = T * M ⊕ R obtained as the sum of the cotangent bundle and of the trivial real line bundle. It can be endowed with a natural Lorentz metric h : for α ∈ T *

x M and u ∈ R, we set h(α, u) = |α| 2 gu 2 . We will say that an element (α, u) of E is future light-like if h(α, u) = 0 and u > 0. We then define a connection ∇ RH on E : if (α, u) is a section of E and X a vector field on M ,

∇ RH X α u := ∇ g X α -ug(X, .) d X u -α(X) .
This connection is metric with respect to h. Moreover, an easy computation shows that its curvature vanishes if and only if g has sectional curvature -1, which is why we call this connection hyperbolic. In case (M, g) is (RH n , g RH n ), this construction is clear : RH n is embedded into Minkowski space R n,1 as the hypersurface Remark 1. Note that this construction admits an obvious spherical analogue. On the same vector bundle, one can consider the obvious positive definite metric and change a sign in the formula for the connection to make it metric, which results in a spherical connection : it is flat if and only if (M, g) is locally isometric to the sphere with constant sectional curvature +1.

x ∈ R n+1 / x 2 1 + • • • + x 2 n -x 2 n+1 = -1 and x n+1 > 0 ; then E = E RH n identifies with T R n,1 | RH n
Observe also that a section (α, u) of E is parallel for this connection if and only if α = du and Hess g u = ug. We call N the space of parallel sections for ∇ RH and N + the subset of future light-like elements of N . Observe that in the model case, N RH n identifies with R n,1 and N + RH n is simply the future isotropic half-cone. Remark 2. The spherical analogue yields the equation Hess g u = -ug, which is called Obata equation and has been much studied ( [Oba, Gal]), in relation with the bottom of the spectrum of Riemannian manifolds with Ric ≥ Ric S n . Obata equation admits a non-trivial solution only on the standard sphere.

We are interested in trying to produce parallel sections of E. To do this, we follow an indirect path, assuming M to be spin and looking at a related spinorial connection. An imaginary Killing spinor ψ on a spin Riemannian manifold (M, g) is a section of the spinor bundle such that

∇ g ψ + i 2 ψ = 0.
The space of Killing spinors is denoted by K. The following crucial observation follows from a straightforward computation :

(4)

∇ g ψ + i 2 ψ = 0 ⇒ ∇ RH d |ψ| 2 g |ψ| 2 g = 0.
In other words, we obtain a map

Q : K -→ N by setting Q(ψ) = (d |ψ| 2 , |ψ| 2 ).
To understand the relevance of imaginary Killing spinors, we must describe them on the model RH n : basically, they are induced by the constant spinors of R n,1 and they trivialize the spinor bundle of RH n [CH]. Using explicit formulas (cf. [CH]), one can see that

N + RH n lies inside Q RH n (K RH n ) ; indeed, it is sufficient to prove that one of the Killing spinors is mapped into N +
RH n and then use the equivariance of Q with respect to the natural actions of O + (n, 1), together with the transitivity of O + (n, 1) on N + RH n . Now assume the complete Riemannian manifold (M, g) is asymptotically hyperbolic in the following sense : M minus a compact is diffeomorphic to RH n minus a ball and, through this diffeomorphism, g = g RH n + O(e -ar ) with a > n+1 2 in C 1,α (r is the distance to some point in RH n ). We further assume that the scalar curvature Scal g of (M, g) is greater than or equal to Scal g RH n . Then [CH] proves that for any Killing spinor ψ on RH n , one can find a unique spinor ψ that is asymptotic to ψ and harmonic for some natural Dirac operator. A Witten's like argument, based on Lichnerowitz formula, then leads to (5)

M ∇ g ψ + i 2 ψ 2 + 1 4 (Scal g -Scal g RH n ) ψ 2 = lim R→∞ S R . . .
The right hand side can be computed explicitly : it is

µ g (Q RH n (ψ)) = - 1 4 lim R-→∞ S R * (div g + d Tr g) |ψ| 2 -Tr(g -g RH n )du + (g -g RH n )(d |ψ| 2 , .) ,
where the sphere, the Hodge star, the divergence, the trace and the identification between vectors and forms on the right-hand side are taken with respect to g RH n .

Since

N + RH n ⊂ Q RH n (K RH n ), it follows that µ yields a linear functional on the linear span of N + RH n , namely on N RH n ∼ = R n,1
. In view of ( 5), it is clearly non-negative on N + RH n . Besides, if µ vanishes, the left-hand side of ( 5) is zero, so that every ψ is a Killing spinor on (M, g) ; then, in view of (4), for any element

σ = Q RH n (ψ) of N + RH n , there is an element σ = Q g ( ψ)
of N g that is asymptotic to σ. As a consequence, N g has maximal dimension, which implies that (M, g) is (locally) hyperbolic. In view of its asymptotic, it is bound to be RH n .

Theorem 1.1 ( [CH]) -Let (M n , g) be an asymptotically hyperbolic spin manifold, with Scal g ≥ Scal g RH n . Then the linear functional µ g on N RH n introduced above is welldefined up to an automorphism of RH n , it is non-negative on N + RH n and it vanishes iff

(M n , g) is isometric to RH n .
The fact that the orbit of µ g under the action of O + (n, 1) does not depend on the chart at infinity is not obvious but is proved in [CH]. Under our assumptions, µ g may take infinite values (i.e. formula (5) may be +∞ for some ψ) and this does not depend on the chart at infinity, so we actually obtain an element of (N RH n ) * /O + (n, 1) ∪ {∞}.

Remark 3. The standard positive mass theorem, about asymptotically Euclidean metrics, can be thought of in a similar way. Let M be a spin asymptotically Euclidean manifold with non-negative scalar curvature. Witten's trick consists in trying to build parallel spinors ψ on M that are asymptotic to the constant spinors ψ of the Euclidean space R n . Now every constant one-form on R n can be written as X → i(X • ψ, ψ). So the rigidity part of the Euclidean positive mass theorem can be explained as follows : starting from a constant one-form α on R n , we pick a constant spinor ψ such that α(X) = (X • ψ, ψ) ; an analytical argument (based on µ g = 0) provides a parallel spinor ψ on M asymptotic to ψ, hence a parallel one-form α, given by α(X) = (X • ψ, ψ), that is asymptotic to α ; this yields a parallel trivialization of the cotangent bundle of M , so M is flat and is therefore R n , owing to its asymptotic shape. The mass is the obstruction to do this. It is a single number µ, but if we wish to make it fit into our picture, we might as well interpret it as a linear functional on the space of parallel sections of the flat bundle

E R n := T * R n ⊕ R : (α, u) → µu.
The bundle E M := T * M ⊕ R, endowed with the Levi-Civita connection on T * M and the flat connection on the R-part, is of course flat if and only if M is flat, so the formalism described above still works.

A complex hyperbolic connection.

2.1. The connection. Let (M m , g, J) be a Kähler manifold of complex dimension m. We wish to introduce a "complex hyperbolic connection" characterizing the complex hyperbolic geometry, in complete analogy with the real hyperbolic connection ∇ RH described in section 1. This is given by the following definition. The Kähler form is denoted by Ω := g(., J.). We will often identify vectors and covectors thanks to the metric g. With our convention, if (e 1 , Je 1 , . . . , e m , Je m ) is an orthonormal basis, then

Ω = m k=1 Je k ∧ e k . Definition 2.1 -Let E := Λ 2
J M ⊕ T * M ⊕ R be the vector bundle obtained as the direct sum of the bundle Λ 2 J M of J-invariant 2-forms, of the cotangent bundle and of the trivial (real) line bundle. We endow E with the connection ∇ CH defined by

∇ CH X   ξ α u   :=   ∇ g X ξ + 1 2 (X ∧ α + JX ∧ Jα) ∇ g X α + 2ι X (ξ + uΩ) d X u + Jα(X)   .
The connection ∇ CH preserves a pseudo-Riemannian structure h on E, with signature

(m 2 + 1, 2m) and given by h(ξ, α, u) = |ξ| 2 g + u 2 - |α| 2 g 2 .
This connection ∇ CH is very related to the study of Hamiltonian two-forms in [ACG], where a similar but more sophisticated connection is introduced. Let us explain why this connection is natural. In analogy with section 1, we expect its curvature to measure the deviation from the complex hyperbolic geometry (in the spirit of Cartan's connections). In order to write down an explicit formula for the curvature, we introduce an algebraic operation : if X and Y are two vectors and γ is an exterior form, we set:

C X,Y (γ) := (X ∧ ι Y γ -Y ∧ ι X γ) + (JX ∧ ι JY γ -JY ∧ ι JX γ).
Proposition 2.2 -The curvature of ∇ CH is given by

Rm ∇ CH X,Y =   Rm ∇ X,Y -C X,Y 0 0 0 Rm ∇ X,Y -[2Ω(X, Y )J + C X,Y ] 0 0 0 0   .
It follows that Rm ∇ CH vanishes if and only if the holomorphic sectional curvature is -4. In other words, (EM, ∇ CH ) is flat if and only if the universal cover of (M m , g, J)

is the complex hyperbolic space CH m of holomorphic sectional curvature -4.

The proof of this formula is a straightforward computation, which we omit. The link with the curvature of the complex hyperbolic space is explained in paragraph IX.7 of [KN] (where the sign convention is the opposite of ours).

Remark 4. By changing ∇ g X α + 2ι X (ξ + uΩ) into ∇ g X α -2cι X (ξ + uΩ)
in the formula for the connection, it is possible to obtain a family of connections characterizing every constant holomorphic curvature 4c (in particular : complex projective spaces).

The parallel sections (ξ, α, u) for ∇ CH obey α = Jdu and ξ = -1 2 ∇ g α -uΩ so that they are determined by their third component u, which satisfies the third order equation

∀ X ∈ T M, ∇ g X Hess g u = 2du(X) + X ⊙ du + JX ⊙ Jdu, where a ⊙ b means a ⊗ b + b ⊗ a.
We will denote by N the space of parallel sections of (E, ∇ CH ). We will also need the subspace ( 6)

N 0 := {(ξ, α, u) ∈ N / g(ξ, Ω) = u}
whose relevance will be clear from paragraph 2.2.

Remark 5. The complex projective analogue (c = +1 in remark 4) of this third order equation appears in Obata's work [Oba] : on a simply-connected manifold, it possesses a solution if and only if the manifold is the complex projective space of holomorphic sectional curvature 4.

2.2. The model case. We wish to describe the case where M = CH m . We start with the complex vector space C m+1 , m ≥ 2, endowed with the Hermitian form <, > defined by

< z, z >= m k=1 |z k | 2 -|z m+1 | 2 .
We denote this space by C m,1 or R 2m,2 (whose metric structure is preserved by the standard complex structure J). The level set < z, z >= -1, endowed with the restriction of <, > is by definition the Anti-de-Sitter space, AdS 2m,1 , a Lorentz manifold with constant sectional curvature -1, invariant under the natural (scalar) action of S 1 on C m+1 . The complex hyperbolic space CH m is the quotient AdS 2m,1 /S 1 , endowed with the induced metric :

AdS 2m,1 ⊂ C m,1 π ↓ CH m
Let ν be the position vector field in C m+1 , identified also with the dual one-form < ν, . >. By definition of AdS 2m,1 , < ν, ν >= -1 along AdS 2m,1 and the tangent bundle T AdS 2m,1 is exactly the orthogonal subspace to ν for <, >. Besides, the vector field Jν is tangent to the action of S 1 and obeys < Jν, Jν >= -1. If π is the projection of AdS 2m,1 onto CH m , it follows that at each point of AdS 2m,1 , dπ is an isometry between {ν, Jν} ⊥ and the tangent space of CH m .

To understand E = E(CH m ), we pick a point z in AdS 2m,1 and look at the map

θ z : E π(z) -→ Λ 2 J R 2m,2 z given by θ z (ξ, α, u) = (dπ z ) * ξ + u(π(z)) Jν ∧ ν + ν ∧ (dπ z ) * α + Jν ∧ J(dπ z ) * α 2 .
It is a R-linear isomorphism and we have

< θ z (ξ, α, u), θ z (ξ, α, u) >= |ξ| 2 π(z) + u(π(z)) 2 - 1 2 |α| 2 π(z) = h π(z) (ξ, α, ν).
Setting Ψ z := θ -1 z , we obtain a bundle map Ψ that is S 1 -invariant and satisfies the commutative diagram

Λ 2 J R 2m,2 Ψ ----→ E     AdS 2m,1 π ----→ CH m
In this way, in the case of CH m , the metric h preserved by ∇ CH simply comes from <, >, via Ψ. At the level of sections, Ψ yields an isomorphism

Γ(E) ∼ = Γ Λ 2 J R 2m,2 | AdS 2m,1 S 1
where the right-hand side denotes the S 1 -invariant sections of the bundle obtained by restricting Λ 2 J R 2m,2 to AdS 2m,1 . This isomorphism identifies σ = (ξ, α, u) ∈ Γ(E) with the S 1 -invariant 2-form

Ψ * σ = π * ξ + π * u Jν ∧ ν + ν ∧ π * α + Jν ∧ Jπ * α 2 .
The bundle Λ 2 J R 2m,2 | AdS 2m,1 carries a natural connection D, inherited from the standard flat connection on R 2m,2 . The real and imaginary parts of the differential forms dz k ∧dz l , restricted to AdS 2m,1 , trivialize the bundle Λ 2 J R 2m,2 | AdS 2m,1 , are S 1 -invariant and Dparallel. D therefore induces, via Ψ, a flat connection on E and the reader might expect the following result.

Proposition 2.3 -The morphism Ψ identifies D and ∇ CH : for every horizontal vector field H on AdS 2m,1 and for every section σ of E, ∇ CH π * H σ = Ψ * D H Ψ * σ. In this statement, "horizontal" means "orthogonal to both ν and Jν". The proof of this is a direct computation, involving only two obvious facts : Dν is the identity at each point and the Levi-Civita connection of the hyperbolic space is induced by D.

Note it is very important to work here with J-invariant 2-forms. For instance, there is no flat S 1 -invariant trivialization of the bundle Λ 1 R 2m,2 | AdS 2m,1 .

We therefore obtain an isomorphism

N CH m ∼ = Λ 2 J R 2m,2 . Let Λ 2 J,0 R 2m,2
be the subspace of primitive J-invariant 2-forms on R 2m,2 . Since the Kähler form ω of C m,1 , restricted to AdS 2m,1 , is given by ω = -Jν ∧ ν + π * Ω, where Ω is the Kähler form of CH m , we also have (cf. ( 6)):

N CH m ,0 ∼ = Λ 2 J,0 R 2m,2 .
2.3. Kählerian Killing spinors. Let (M m , g, J) be a spin Kähler manifold. As in the hyperbolic space, a class of special spinors somehow characterizes the complex hyperbolic space. It turns out that there is a dimension issue, so we first assume the complex dimension m is odd : m = 2l -1. The even dimensional case will be discussed in section 4.

The spinor bundle Σ decomposes into the orthogonal sum of the eigenspaces of the natural action of the Kähler form Ω = g(., J.) : Σ = m k=0 Σ k , where Σ k corresponds to the eigenvalue i(m -2k) ( [Kir] for instance). We will write π Ω k (ψ) or simply ψ k for the kth component of the spinor ψ in this decomposition. From another point of view, spinors may be seen as twisted forms : Σ = m k=0 Λ 0,k ⊗ √ Λ m,0 and then

Σ k = Λ 0,k ⊗ √ Λ m,0
. Through this identification, Clifford product "•" is merely √ 2 times the difference between exterior product and interior product ((1, 0)-vectors are identified with (0, 1)-covectors by the Hermitian inner product). As a consequence, (7)

T 1,0 • Σ k ⊂ Σ k+1 and T 0,1 • Σ k ⊂ Σ k-1 .
The following formula defines a connection ∇ on the vector bundle Σ :

∇X ψ = ∇ X ψ + ic(X 1,0 )ψ l-1 + ic(X 0,1 )ψ l = ∇ X ψ + i 2 c(X)(ψ l-1 + ψ l ) + 1 2 c(JX)(ψ l-1 -ψ l ).
The sections of Σ l-1 ⊕ Σ l that are parallel for the connection ∇ are called imaginary Kählerian Killing spinors. The dimension of the space K of Kählerian Killing spinors is therefore at most C l 2l . This bound is attained on CH m , as we will see in details after corollary 2.7. First, we make a number of general useful remarks.

For future reference, let us introduce the (modified) Dirac operator ˆ D which is naturally associated with the connection ∇ : if (ǫ j ) j is any g-orthonormal frame, it is given by ˆ Dψ := 2m j=1 c(ǫ j ) ∇ǫ j ψ.

In [Her], another modified Dirac operator D is used, for analytical reasons :

Dψ := ˆ Dψ -i(m + 1)(ψ -ψ l-1 -ψ l ) = D ψ -i(m + 1)ψ,
where D is the standard Dirac operator. It should be noticed that Kählerian Killing spinors are canceled by both ˆ D and D.

Given a spinor φ = φ l-1 + φ l ∈ Σ l-1 ⊕ Σ l , we will use the notation φ := φ l-1φ l ; φ is (-1) l+1 times the conjugate φ +φ -with respect to the usual decomposition into half-spinors. The Kählerian Killing equation can then be written

∇ X φ = - i 2 X • φ - 1 2 (JX) • φ or ∇ X φ = i 2 X • φ + 1 2 (JX) • φ.
The following lemma is easy but very useful.

Lemma 2.4 -If φ is a Kählerian Killing spinor then

(X • φ, φ) = -X • φ, φ = 2i Im X 1,0 • φ l-1 , φ l (X • φ, φ) = -X • φ, φ = 2 Re X 1,0 • φ l-1 , φ l .
In particular, ((JX) • φ, φ) = i(X • φ, φ).

Proof. The first statement follows from the computation

(X • φ, φ) = (X • (φ l + φ l-1 ), φ l + φ l-1 ) = (X • φ l , φ l-1 ) + (X • φ l-1 , φ l ) = 2i Im (X • φ l-1 , φ l ) = 2i Im X 1,0 • φ l-1 , φ l
and the other ones are similar.

In complete analogy with the real hyperbolic case, the squared norm |φ| 2 of a Kählerian Killing spinor φ will induce a ∇ CH -parallel section of E. To see this, we first compute two derivatives of this function.

Lemma 2.5 -Any Kählerian Killing spinor φ obeys

d |φ| 2 (X) = -2i(X • φ, φ) (8) ∇ X d |φ| 2 (Y ) = 2(X, Y ) |φ| 2 -2 Im Y • (JX) • φ, φ . (9)
Proof. The definition of Kählerian Killing spinors readily yields

d |φ| 2 (X) = - i 2 (X • φ, φ) - 1 2 (JX • φ, φ) + i 2 (φ, X • φ) - 1 2 (φ, JX • φ),
which leads to d |φ| 2 (X) = -2i(X • φ, φ), thanks to lemma 2.4. Differentiating once more, we find

∇ X d |φ| 2 (Y ) = -2i(Y • ∇ X φ, φ) -2i(Y • φ, ∇ X φ) = -(Y • X • φ, φ) + i(Y • JX • φ, φ) + (Y • φ, X • φ) + i(Y • φ, JX • φ)
and the result follows from elementary properties of the Clifford product.

At this point, it is useful to introduce the following notations : for every Kählerian Killing spinor φ, we define :

u φ := |φ| 2 , α φ := Jdu φ , ξ φ (X, Y ) := Im(X • Y • φ, φ).
Lemma 2.6 -ξ φ is a J-invariant two-form satisfying :

∇ Z α φ = -2ι Z (ξ φ + u φ Ω), (ξ φ , Ω) = u φ , ∇ Z ξ φ = - 1 2 (Z ∧ α φ + JZ ∧ Jα φ ) .
Proof. First, it is easy to check that ξ φ is skewsymmetric, as a consequence of the definition of φ. The J-invariance of ξ φ stems from the following reformulation of its definition :

ξ φ (X, Y ) := Im(X 1,0 • Y 0,1 • φ, φ) + Im(X 0,1 • Y 1,0 • φ, φ).
Then formula (9) readily yields

∇ X α φ (Y ) = -2(X, JY )u φ + 2ξ φ (JY, JX) = -2u φ Ω(X, Y ) -2ξ φ (X, Y ),
which justifies the first formula.

Let (e 1 , Je 1 , . . . , e m , Je m ) be an orthonormal basis. Then we have

(ξ φ , Ω) = m k=1 ξ φ (Je k , e k ) = m k=1 Im(Je k • e k • φ, φ) = Im(Ω • φ, φ).
Now we use the spectral decomposition of the action of the Kähler form :

(Ω • φ, φ) = (Ω • φ l-1 -Ω • φ l , φ) = (iφ l-1 + iφ l , φ) = i |φ| 2 . This ensures (ξ φ , Ω) = |φ| 2 = u φ .
Finally, to obtain the third equation, we introduce θ(X, Y ) := 2(X • Y • φ, φ) and differentiate :

∇ Z θ(X, Y ) = 2(X • Y • ∇ Z φ, φ) + 2(X • Y • φ, ∇ Z φ) = i(X • Y • Z • φ, φ) + (X • Y • (JZ) • φ, φ) + i(X • Y • φ, Z • φ) -(X • Y • φ, (JZ) • φ) = i([X • Y • Z -Z • X • Y ] • φ, φ) + ([X • Y • (JZ) -(JZ) • X • Y ] • φ, φ). The identity ABC -CAB = 2(A, C)B -2(B, C)A in Clifford algebra leads to ∇ Z θ(X, Y ) = 2i(X, Z)(Y • φ, φ) -2i(Y, Z)(X • φ, φ) + 2(X, JZ)(Y • φ, φ) -2(Y, JZ)(X • φ, φ).
Lemmata 2.4 and 2.5 then yield :

∇ Z θ(X, Y ) = i(X, Z)du φ (JY ) -i(Y, Z)du φ (JX) + i(X, JZ)du φ (Y ) -i(Y, JZ)du φ (X) = -i(X, Z)α φ (Y ) + i(Y, Z)α φ (X) -i(X, JZ)Jα φ (Y ) + i(Y, JZ)Jα φ (X)
and the result follows.

The computations above result in the following proposition (recall( 6)).

Corollary 2.7 -If φ is a Kählerian Killing spinor, then Q(φ) := (ξ φ , α φ , u φ ) is an element of N 0 . So we have a map Q : K -→ N 0 .
When m = 2l -1 for some integer l ≥ 2, the complex hyperbolic space CH m is known to carry a space of Kählerian Killing spinors of maximal dimension, C l 2l [Kir]. Where do they come from ? Briefly, constant spinors on C m,1 admit a restriction as spinors on AdS 2m,1 (cf. Lemma 3 in [Bau]). These are the so-called imaginary Killing spinors ; they trivialize the spinor bundle of AdS 2m,1 . Among them, thanks to the parity of m, some are S 1 -invariant (it can be seen on the graduation coming from the Kähler structure of C m,1 ) and admit a projection into spinors along CH m ( [Mor]). These "projected" spinors are exactly the Kählerian Killing spinors.

To be more explicit, we can adapt the computations of [Kir]. The manifold CH m carries global coordinates w 1 , . . . , w m , such that π * w k = z k z m+1 (along AdS 2m,1 , where z m+1 does not vanish). These coordinates induce a trivialization of the canonical bundle, by [Kir] say that, if a is a multi-index of length l -1, the spinor ϕ a = ϕ a l-1 + ϕ a l defined by

√ dw := √ dw 1 ∧ • • • ∧ dw m . We will use multi-index a = (a 1 , . . . , a k ) with 1 ≤ a 1 < • • • < a k ≤ m and set dz a := dz a 1 ∧ • • • ∧ dz a k . The computations of
ϕ a l-1 = c(l) d wa (1 -|w| 2 ) l ⊗ √ dw and ϕ a l = c(l) 2il ∂ d wa (1 -|w| 2 ) l ⊗ √ dw
is a Kählerian Killing spinor. In this expression, c(l) is a normalization constant, which we choose to be c(l) = √ 2 5 2 -3l . Another family of Kählerian Killing spinors is described by spinors φb = φb l-1 + φb l where b is a multi-index of length l and

φb l-1 = c(l) ι Rd wb (1 -|w| 2 ) l ⊗ √ dw and φb l = c(l) 2il ∂ ι Rd wb (1 -|w| 2 ) l ⊗ √ dw where R = k w k ∂ w k .
These families together form a basis for the Kählerian Killing spinors of CH m . What we are interested in here is their squared norms. Let us introduce the notation ȃ for the multi-index that is complementary to a (namely, a and ȃ have no common index and the sum of their lengths is m), and also the notation

|w a | 2 := |w a 1 | 2 + • • • + |w a k | 2 .
An adaptation of the computations at the end of [Kir] yields

ϕ a l-1 2 = | φȃ l | 2 = 1 -|w a | 2 1 -|w| 2 and |ϕ a l | 2 = | φȃ l-1 | 2 = |w ȃ| 2 1 -|w| 2 , hence |ϕ a | 2 = | φȃ | 2 = 1 -|w a | 2 + |w ȃ| 2 1 -|w| 2 .
It follows that π * u ϕ a = -|z a | 2 + |z ȃ| 2 + |z m+1 | 2 (we forget the φ's since they do not yield new squared norms). Now α ϕ a and ξ ϕ a are determined by u ϕ a (since these are the components of an element of N ). Denoting the real part of z k by x k , one can check that the corresponding element of

N 0 (CH m ) ∼ = Λ 2 J,0 R 2m,2 is β ϕ a = - l-1 j=1 Jdx a j ∧ dx a j + l j=1 Jdx ȃj ∧ dx ȃj + Jdx m+1 ∧ dx m+1
(To see this, it is sufficient to compute the scalar product of the right-hand side with Jν ∧ ν and observe that it coincides with π * u ϕ a .) In particular, < β ϕ a , β ϕ a >= m + 1. All these two-forms belong to the same orbit of the isometric and holomorphic action of P U (m, 1). We will denote this orbit by N + 0,CH m . Note that, since Λ 2 J,0 R 2m,2 is an irreducible representation of P U (m, 1), the linear span of N + 0,CH m is the whole Λ 2 J,0 R 2m,2 .

3. Toward a mass.

3.1.

The "mass integral" at infinity. Let us give a precise definition for the class of manifolds we are interested in.

Definition 3.1 -A complete Kähler manifold (M m , g, J) is called asymptotically complex hyperbolic if there is a compact subset K of M and a ball B in CH m such that:

(i) (M \K, J) is biholomorphic to CH m \B and, (ii) through this identification, e ar (gg CH m ) is bounded in C 1,α , with respect to the complex hyperbolic metric g CH m . Here, r denotes the distance to some point in CH m and we assume a > m + 1 2 . In [Her], it is only assumed that the complex structure J of M is asymptotic to the complex structure J 0 of CH m (instead of J = J 0 , as in our definition). It turns out that, if J is asymptotic to J 0 , then they are related by a biholomorphism, defined outside a compact set and asymptotic to the identity. To justify this, let us see J and J 0 as two complex structures on a neighborhood U of S 2m-1 in the unit ball of C m . Under our assumptions, they induce the same CR structure on the unit sphere, the standard CR structure of S 2m-1 . The restriction of any J 0 -holomorphic coordinate z k to the sphere is a CR function. Since the standard sphere is strictly-pseudo-convex, Lewy's extension theorem makes it possible to extend this function into a J-holomorphic function w k on U (shrinking U if necessary). This yields a holomorphic map w = (w 1 , . . . , w m ) from Ω to C m . Since z = (z 1 , . . . , z m ) is a diffeomorphism onto its range and coincide with w on the sphere, we may shrink U to ensure w is also a diffeomorphism onto its range, hence a biholomorphism between two neighborhoods of S 2m-1 in the unit ball of C m , one endowed with J and the other one with the standard complex structure. The promised biholomorphism is z -1 • w.

In this section, we assume (M, g, J) is an asymptotically complex hyperbolic manifold of odd complex dimension m = 2l -1 and with scalar curvature bounded from below by -4m(m + 1) (the scalar curvature of CH m ). In this setting, the spinor bundle Σ| M \K can be identified with the spinor bundle of CH m \B. We may therefore extend any Kählerian Killing spinor φ on CH m \B into a spinor φ on M . Our aim is to understand to what extent we can make it into a Kählerian Killing spinor. In the spirit of [Wit], we first check that we can choose an extension in the kernel of a Dirac operator.

Lemma 3.2 -There is a smooth spinor ψ := φ + φ err such that Dψ = 0 and φ err decays as e -br in C 1,α with b > m + 1 2 .

Proof. We first observe that

DD * = D 2 +(m + 1) 2 = ∇ * ∇ + 1 4 Scal +(m + 1) 2 ≥ ∇ * ∇ + m + 1.
A slight adaptation of Proposition I.3.5 in [Biq] then shows that the operator

DD * : e br C 2,α -→ e br C 0,α is an isomorphism for every b such that m - √ m 2 + m + 1 < b < m + √ m 2 + m + 1. Since Dφ = (D g -D CH m )φ = O(e -(a-1)r
), with a > m+ 1 2 , we can pick a b > m+ 1 2 such that the equation DD * σ = -Dφ 0 admits a solution σ in e br C 2,α . Then φ err := D * σ is convenient.

We then invoke a Weitzenböck formula, proved in paragraph 3 of [Her] (modulo two misprints, indeed: in the formula stated, an i should be added at the second and third lines and the coefficient m -2q and the fourth one should be replaced by 2(mq)) :

S R * ζ ψ,ψ = B R ∇ψ 2 + 1 4 B R (Scal +4m(m + 1)) |ψ| 2 + (m + 1) B R |ψ| 2 -π Ω l-1 ψ 2 -π Ω l ψ 2 ,
where S R denotes the sphere {r = R}, bounding the domain B R , and

ζ σ,τ is the 1-form defined by ζ σ,τ (X) = ( ∇X σ + c(X)Dσ, τ ).
In view of this formula, the obstruction for ψ to be a Kählerian Killing spinor is precisely the "mass integral at infinity" lim R→∞ S R * ζ ψ,ψ , which is a well defined element of [0, +∞], because the integrand on the right-hand side is non-negative.

Lemma 3.3 -lim R→∞ S R * ζ ψ,ψ = lim R→∞ S R * ζ φ,φ .
Proof. Since Re ζ σ,τ is symmetric up to a divergence term (as noticed in [Her], p.651), we only need to check that lim

R→∞ S R * (ζ φ,φerr + ζ φerr,φerr ) = 0.
This follows from the following estimates : vol S R = O(e 2mr ), ∇φ = O(e (1-a)r ) (beware φ grows in e r ), φ err = O(e -br ), ∇φ err = O(e -br ) (cf. lemma 3.2), with a > m + 1 2 and b > m + 1 2 .

In order to compare the metrics g et g 0 := g CH m , we introduce the symmetric endomorphism A such that g 0 = g(A., A.). Since A maps g 0 -orthonormal frames to g-orthonormal frames, it identifies the spinor bundles defined with g and g 0 (cf. [CH] for instance). The associated Clifford products c g and c g 0 are related by the formula c g (AX)σ = c g 0 (X)σ. Note we will also write X• for c g 0 (X) (and not for c g (X)). For the sake of efficiency, we will write u ≈ v when uv = o(e -2mr ) ; the terms we neglect in this way will indeed not contribute to the integral at infinity. Before computing the "mass integral at infinity", we point out a few elementary facts.

Lemma 3.4 -A -1 JA ≈ J .
Proof. The definition g 0 = g(A, A) and the compatibility of J with g 0 yield the equality g(AJ, AJ) = g(A, A). Since A and J are respectively g-symmetric and g-antisymmetric, we deduce : JA 2 J = -A 2 . If A = 1 + H, this implies JHJ ≈ -H. Since J 2 = -1, we obtain JH ≈ HJ and then JA ≈ AJ, hence the result.

Corollary 3.5 -We have c g (Ω) ≈ c g 0 (Ω 0 ) and π Ω k ≈ π Ω 0 k . Proof. Given a g 0 -orthonormal basis (e 1 , Je 1 , . . . , e m , Je m ), the Clifford action of the Kähler form Ω 0 reads c g 0 (Ω 0 ) = k Je k • e k • while the Kähler form Ω of g acts by

c g (Ω) = m k=1 c g (JAe k )c g (Ae k ) = m k=1 (A -1 JAe k ) • e k • .
The first statement is therefore a straightforward consequence of lemma 3.4. The second one follows from general considerations. We observe the skew-Hermitian endomorphisms P := c g (Ω) and P 0 := c g 0 (Ω 0 ) act on each fiber of the spinor bundle with the same spectrum. If λ is one of the eigenvalues, the corresponding spectral projectors Π and Π 0 (for P and P 0 ) obey the formulas

Π = 1 2π C (z -P ) -1 dz and Π 0 = 1 2π C (z -P 0 ) -1 dz
where C is a circle in the complex plane, centered in λ and with small radius δ. We deduce

Π -Π 0 = 1 2π C (z -P ) -1 (P -P 0 )(z -P 0 ) -1 dz and then |Π -Π 0 | ≤ δδ -1 |P -P 0 | δ -1 = δ -1 |P -P 0 | .
The result follows at once.

The rest of this section is devoted to the proof of the following statement.

Proposition 3.6 -The "mass integral at infinity" is

lim R→∞ S R * ζ ψ,ψ = lim R→∞ S R * - 1 4 (d Tr g 0 g + div g 0 g) |φ| 2 + 1 8 Tr g 0 (g -g 0 ) d |φ| 2 .
Proof. To begin with, in view of corollary 3.5, we may write

ζ φ,φ (Y ) = ( ∇g Y φ + c g (Y ) ˆ Dφ, φ) -i(m + 1) c g (Y )(1 -π Ω l-1 -π Ω l )φ, φ ≈ ( ∇g Y φ + c g (Y ) ˆ Dφ, φ).
Given a g 0 -orthonormal frame (e 1 , . . . , e 2m ) and a g 0 -unit vector X, since φ is a Kählerian Killing spinor with respect to (g 0 , J 0 ), we may therefore write outside K (as in [CH, Min] for instance) :

ζ φ,φ (AX) ≈ 1 2 2m j=1 ([c g (AX), c g (Ae j )] ∇g Ae j φ, φ) ≈ 1 2 2m j=1 ([c g (AX), c g (Ae j )]( ∇g Ae j -∇g 0 Ae j )φ, φ) ≈ 1 2 2m j=1 ([X•, e j •]( ∇g Ae j -∇g 0 Ae j )φ, φ), which expands into 1 2 2m j=1 [X•, e j •](∇ g Ae j -∇ g 0 Ae j )φ, φ + i 4 2m j=1 [X•, e j •] c g (Ae j -iJAe j )π Ω l-1 -c g 0 (Ae j -iJAe j )π Ω 0 l-1 φ, φ + i 4 2m j=1 ([X•, e j •] c g (Ae j + iJAe j )π Ω l -c g 0 (Ae j + iJAe j )π Ω 0 l φ, φ , that is 1 2 2m j=1 [X•, e j •](∇ g Ae j -∇ g 0 Ae j )φ, φ + i 4 2m j=1 [X•, e j •] (e j -iA -1 JAe j ) • π Ω l-1 -(Ae j -iJAe j ) • π Ω 0 l-1 φ, φ + i 4 2m j=1 ([X•, e j •] (e j + iA -1 JAe j ) • π Ω l -(Ae j + iJAe j ) • π Ω 0 l φ, φ .
In view of lemma 3.4 and corollary 3.5, we are left with :

ζ φ,φ (AX) ≈ 1 2 2m j=1 [X•, e j •](∇ g Ae j -∇ g 0 Ae j )φ, φ + i 4 2m j=1 [X•, e j •]((e j -iJe j ) -(Ae j -iJAe j )) • π Ω 0 l-1 φ, φ + i 4 2m j=1 [X•, e j •]((e j + iAe j ) -(Ae j + iJAe j )) • π Ω 0 l φ, φ .
With A = 1 + H we can there therefore write ζ φ,φ (AX) ≈ I + II + III, with :

I := 1 2 2m j=1 [X•, e j •](∇ g Ae j -∇ g 0 Ae j )φ, φ II := - i 4 2m j=1 ([X•, e j •]He j • φ, φ) III := - 1 4 2m j=1 [X•, e j •]JHe j • φ, φ .
The computation of the real part of the first term is classical (cf. [CH] or lemma 10 in [Min], for instance) :

Re I ≈ - 1 4 (d Tr g 0 g + div g 0 g) |φ| 2 .
The second term is basically computed in [CH]. Indeed, since H is symmetric, we use the identity [X•, e j •] = 2X j + 2X • e j • to obtain

Re II = - i 2 (HX • φ, φ) - i 2 2m j,l=1 H jl (X • e j • e l • φ, φ) = - i 2 (HX • φ, φ) + i 2 Tr H (X • φ, φ) .
In the same way, the third term can be written

III = - 1 2 (JHX • φ, φ - 1 2 2m j=1 X • e j • JHe j • φ, φ . Lemma 3.7 -Re 2m j=1 X • e j • JHe j • φ, φ ≈ -2 JHX • φ, φ .
Proof. Let us set M := JH and M ij := (e i , M e j ), so that

2m j=1 X • e j • JHe j • φ, φ = j,k,p M kj X p e p • e j • e k • φ, φ .
Lemma 3.4 ensures JH ≈ HJ. Since H is symmetric and J antisymmetric, we deduce that M is antisymmetric up to a negligible term. In particular, M kj ≈ 0 when k = j, hence and since the indices are distinct, this is imaginary.

So Re 2m j=1 X • e j • JHe j • φ, φ ≈ Re j =k M kj X j e j • e j • e k • φ, φ + Re j =k M kj X k e k • e j • e k • φ, φ ≈ -Re j =k M kj X j e k • φ, φ + Re j =k M kj X k e j • φ, φ ≈ -2 Re j =k M kj X j e k • φ, φ ≈ -2 Re M X • φ, φ ≈ -2 M X • φ, φ .
spinor in the chart Ψ 1 , then φ 2 := f * φ 1 is a Kählerian Killing spinor in the chart Ψ 2 (by naturality). We claim that

Lemma 3.8 -µ Ψ 1 g (β φ 1 ) = µ Ψ 2 g (β φ 2 ). Proof.
Recall that in the notations introduced in the previous paragraph,

µ Ψ 1 g (β φ 1 ) = lim R→∞ S Ψ 1 R * ζ φ 1 ,φ 1 = lim R→∞ S Ψ 1 R * ζ ψ 1 ,ψ 1 .
This quantity is equal to the well-defined integral

M ∇ψ 1 2 + 1 4 M (Scal +4m(m + 1)) |ψ 1 | 2 +(m + 1) M |ψ 1 | 2 -π Ω l-1 ψ 1 2 -π Ω l ψ 1 2 ,
so that we can compute it with another family of spheres :

µ Ψ 1 g (β φ 1 ) = lim R→∞ f (S Ψ 1 R ) * ζ ψ 1 ,ψ 1 = lim R→∞ f (S Ψ 1 R ) * ζ φ 1 ,φ 1 . Since f * g 0 = g 0 and f (S Ψ 1 R ) = S Ψ 2 R , we therefore obtain µ Ψ 1 g (β φ 1 ) = lim R→∞ f (S Ψ 1 R ) * g 0 - 1 4 (d Tr g 0 g + div g 0 g) |φ 1 | 2 g 0 + 1 8 Tr g 0 (g -g 0 ) d |φ 1 | 2 g 0 = lim R→∞ S Ψ 2 R * g 0 - 1 4 (d Tr g 0 g + div g 0 g) |f * φ 2 | 2 g 0 + 1 8 Tr g 0 (g -g 0 ) d |f * φ 2 | 2 g 0 . = lim R→∞ S Ψ 2 R * g 0 - 1 4 (d Tr g 0 g + div g 0 g) |φ 2 | 2 g 0 + 1 8 Tr g 0 (g -g 0 ) d |φ 2 | 2 g 0 = µ Ψ 2 g (β φ 2 ).
In other words, changing the chart at infinity by an automorphism f ∈ P U (m, 1) results in turning µ g to f * µ g . So µ g is well-defined up to the natural action of P U (m, 1). We have proved the following result.

Theorem 3.9 -Let (M m , g, J) be a spin asymptotically complex hyperbolic Kähler manifold with odd complex dimension Scal g ≥ Scal CH m . Then the linear functional µ g on N CH m ,0 is well-defined up to the natural action of P U (m, 1) ; it is non-negative on N + CH m ,0 and vanishes if and only if (M m , g, J) is the complex hyperbolic space.

4. The case of even-dimensional manifolds.

4.1. The twisted Kählerian Killing spinors. To extend the ideas above to the evendimensional case, it seems that we need Kählerian Killing spinors on even-dimensional complex hyperbolic spaces. Unfortunately, such Kählerian Killing spinors do not exist.

To overcome this cruel reality, we follow [BH] and turn to the spin c realm. We refer to [LM] for basic definitions about spin c structures. We consider a Kähler manifold (M, g, J) with even complex dimension m = 2l. As in [BH], we further assume that the cohomology class of Ω iπ is integral, i.e. in the image of H 2 (M, Z) → H 2 (M, R). This determines a complex line bundle L endowed with a Hermitian metric and a unitary connection with curvature F = -2iΩ ; the Chern class of

L is c 1 (L) = i 2π [F ] = 1 π [Ω]
. We also assume that (M, L) defines a spin c structure, in that the bundle T M ⊗ L admits a spin structure. We then introduce the corresponding spinor bundle Σ c , endowed with a Clifford action c and a connection ∇. The Kähler form Ω acts on this bundle, with the eigenvalues i(m-2k), 0 ≤ k ≤ m. The eigenspaces yield subbundles Σ c k . We may therefore define a connection ∇ on Σ c by requiring that ∇X

ψ := ∇ X ψ + ic(X 1,0 )ψ l-1 + ic(X 0,1 )ψ l ,
where ψ k is the component of ψ in Σ c k . The parallel sections of Σ c l-1 ⊕ Σ c l for this connection will be called twisted Kählerian Killing spinors and the corresponding subspace will be denoted by K c .

Every computation of section 2.3 (but one) can be carried out with twisted Kählerian Killing spinors, leading to the same formulas. In particular, one can define u ψ , α ψ , ξ ψ for any ψ in K c as in section 2.3 and this yields a map Q c : K c → N . The only difference is that we do not get elements of N 0 : (ξ ψ , Ω) is no longer u ψ , basically because the eigenvalues of the Kähler form are now even (compare with the proof of Lemma 2.6).

Let us describe the model case, where (M, g, J) is the complex hyperbolic space CH m , m = 2l, with holomorphic sectional curvature -4. With this normalization, we have Ric CH m = -2(m + 1)g CH m . Since the Ricci form ρ = Ric(., J.) is i times the curvature of the canonical line bundle, this implies :

c 1 (Λ m,0 ) = i 2π [-iρ] = i 2π [2i(m + 1)Ω] = - m + 1 π [Ω] = -(m + 1)c 1 (L).
So L ∼ = Λ m,0 -1 m+1 in this case. It follows that

Σ c k ∼ = Σ k ⊗ L ∼ = Λ 0,k ⊗ Λ m,0 1 2 -1 m+1 ∼ = Λ 0,k ⊗ Λ m,0 l 2l+1 .
It turns out that the sections of K c trivialize the bundle Σ c l-1 ⊕ Σ c l , as noticed in [BH]. Indeed, using the same notations as in section 2.3, we can define two families of twisted Kählerian Killing spinors in the following way. First, if a is a multi-index of length l -1, we set ϕ a = ϕ a l-1 + ϕ a l with ϕ a l-1 = c(l) d wa (1 -|w| 2 ) l ⊗ dw where R = k w k ∂ w k as before. These families together form a basis for K c on CH m . One can again compute the squared norms of these spinors, like in [Kir]. If a is a multi-index of length l -1 and b a multi-index of length l, we have

ϕ a l-1 2 = 1 -|w a | 2 1 -|w| 2 , |ϕ a l | 2 = |w ȃ| 2 1 -|w| 2 , φb l-1 2 = |w b | 2 1 -|w| 2 , φb l 2 = 1 -w b 2 1 -|w| 2 , hence |ϕ a | 2 = 1 -|w a | 2 + |w ȃ| 2 1 -|w| 2 , φb 2 = 1 -w b 2 + |w b | 2 1 -|w| 2 .
map for the standard action of S 1 on C m \ {0} endowed with the symplectic form ω. If I denotes the range of the function ψ, we define the "momentum profile" Θ as the positive function Θ := ψ ′ • ψ -1 on I =] inf ψ, sup ψ[. The Kähler metric extends smoothly near 0 if and only if inf ψ = 0 and Θ is smooth near 0 with Θ(0) = 0 and Θ ′ (0) = 2. Setting x := ψ(t) ∈ I, the Kähler form is given by

ω = ω Θ = dx ∧ d c x Θ(x) + x ω F S
and its scalar curvature is

s Θ (x) = 2m(m -1) x - ∂ xx (x m-1 Θ(x))
x m-1 .

The flat metric corresponds to Θ(x) = 2x, x ∈ R + , the complex hyperbolic metric to Θ(x) = 2x + 2x 2 , x ∈ R + and the Fubini-Study metric on CP m to Θ(x) = 2x -2x 2 , x ∈ [0, 1]. Let Θ 0 (x) = 2x 2 +2x be the momentum profile of the complex hyperbolic metric. We wish to build a U (m)-invariant Kähler metric with complex hyperbolic asymptotic and scalar curvature bounded from below by the scalar curvature of the complex hyperbolic model. We therefore need to find a positive function Θ defined on R + , such that Θ(0) = 0, Θ ′ (0) = 2, Θ(x) is asymptotic to 2x + 2x 2 as x goes to infinity and s Θ ≥ s Θ 0 . Setting Θ(x) = Θ 0 (x)-α(x), we are lead to find a non zero function α such that Θ 0 -α is non-negative, α(0) = α ′ (0) = 0, α = o(x 2 ) as x goes to infinity and ∂ xx (x m-1 α(x)) ≥ 0. Such a function α can be obtained by choosing where χ is a bump function with support inside [1, +∞[ and with unit integral. One can then observe that α(x) vanishes on [0, 1] and is equivalent to x 2-m as x goes to infinity, while satisfying the desired differential inequality ; besides, the normalization of χ ensures α(x) ≤ x for every x ≥ 0, which guarantees Θ 0α ≥ 0. The Kähler form is then

ω Θ = ω Θ 0 + Θ 0 (x) Θ(x) -1 dx ∧ d c x Θ 0 (x)
with Θ 0 (x) Θ(x) -1 ∼ x -m 2 . To understand the asymptotic of this, we relate x to the initial variable t 0 on the complex hyperbolic space. The properties dt 0 dx = 1 Θ 0 (x) and t 0 (x = +∞) = 0 yield t 0 ∼ -1 2x . The geodesic distance r to 0 in the complex hyperbolic model is given by r = tanh e t 0 so that x ∼ e 2r 4 at infinity. So our Kähler metric satisfies ω = ω 0 + O(e -2m r ).

Remark. The rate of the falloff to the model at infinity that we obtained is optimal in the radial case. In other words, a function α satisfying the desired properties cannot decay faster at infinity. This is due to the fact that the function x → x m-1 α(x) must be convex on R + with zero first order jet at 0 ; so either it vanishes identically or it grows at least linearly.

Date:

  November 2, 2009November 2, . 2000 Mathematics Subject Classification. The authors benefit from the ANR grant GeomEinstein, ANR-06-BLAN-0154-02.

  S R * (div g + d Tr g) ,

  , h is induced by the Minkowski metric and ∇ RH by the flat Minkowski connection.

X

  • e j • JHe j • φ, φ ≈ j =k p M kj X p e p • e j • e k • φ, φ . Given three distinct indices j, k, p we consider the expression e p • e j • e k • φ, φ = (e p • e j • e k • (φ l-1φ l ), (φ l-1 + φ l )) . Property (7) reduces it into e p • e j • e k • φ, φ = (e p • e j • e k • φ l-1 , φ l ) -(e p • e j • e k • φ l , φ l-1 ) .

  |w| 2 ) l ⊗ dw l 2l+1 .The normalization we choose is c(l) = 2 1-l-l 2 2l+1 . We also introduce φb = φb l-1 + φb l where b is a multi-index of length l andφb l-1 = c(l) ι Rd wb (1 -|w| 2 ) l ⊗ dw |w| 2 ) l ⊗ dw l 2l+1

This lemma leads to

Re III ≈ -

Eventually, summing I, II and III, we get :

Lemmata 2.4 and 2.5 simplify this into Re ζ φ,φ (AX) ≈ -1 4 (d Tr g 0 g + div g 0 g)

Since gg 0 ≈ -2g 0 (H., .), we have Tr(gg 0 ) ≈ -2 Tr H, hence :

which yields the formula of proposition 3.6.

3.2.

The mass linear functional. We consider the formula

), the considerations of the previous paragraph imply that µ g (ξ, α, u) is an element of [0, +∞]. If it is infinite at some point of Q CH m (K CH m ), we decide that the mass is infinite : µ g = ∞. Otherwise, this defines a linear functional µ g on the linear span of Q CH m (K CH m ) ⊂ N CH m ,0 , which is N CH m ,0 , since it is P U (m, 1)invariant and the action of this group on Λ 2 J,0 R 2m,2 is irreducible. The previous paragraph also indicates that µ g takes non-negative values on (the convex cone generated by)

. Assume µ g vanishes on N CH m ,0 . Then Lemma 3.2, coupled to the Bochner formula, ensures that for every ψ in K CH m , there is a Kählerian Killing spinor φ on (M, g), namely φ ∈ K g , that is asymptotic to ψ. In particular, for any element

CH m ,0 is the whole N CH m ,0 , we deduce that N g,0 has maximal dimension, which implies that N g has maximal dimension (add the Kähler form) : (M, g) is (locally) complex hyperbolic. In view of its asymptotic, it is bound to be CH m .

The holomorphic chart at infinity in Definition 3.1 is of course not unique. Any two relevant charts Ψ 1 and Ψ 2 differ by a biholomorphism f := Ψ 2 • Ψ -1 1 of CH m such that f * g 0 is asymptotic to g 0 . The map f therefore induces a CR-automorphism of the sphere at infinity, so that f is asymptotic to an element of P U (m, 1). As in the remark after Definition 3.1, we conclude f is an element of P U (m, 1). So the model at infinity is unique up to its natural automorphism group P U (m, 1). If φ 1 is a Kählerian Killing Exactly as in the odd dimensional case, we see that the corresponding elements of

The β ϕ a 's all belong to the same orbit under the action of P U (m, 1) : let N +1 CH m be this orbit. The β φb 's also belong to the same orbit under the action of P U (m, 1) and we denote it by N -1 CH m . 4.2. The positive mass theorem.

Theorem 4.1 -Let (M m , g, J) be an asymptotically complex hyperbolic Kähler manifold with even complex dimension and Scal g ≥ Scal CH m . We assume that the cohomology class of Ω iπ is integral, providing a line bundle L as above, and that (M, L) defines a spin c structure. Then the formula (10) defines a (possibly infinite) linear functional µ g on N CH m . It is well defined by g up to the natural action of P U (m, 1), non-negative on N +1

CH m ∪ N -1 CH m and vanishes if and only if (M m , g, J) is the complex hyperbolic space. Note the assumption about [Ω] and L are of course satisfied when M is contractible.

Proof. The proof is nearly the same as in the odd dimensional case. The analytical part is completely similar, cf. [BH]. So, for every element φ of K c CH m , one may find an harmonic spinor ψ on M that is asymptotic to φ (harmonic means in the kernel of some Dirac operator, cf. [BH]). We may then proceed to the same computation, leading to the same mass integral at infinity, µ g , defined on the linear span of Q c (K c ) in N . From the computations above we know that Q c (K c ) contains N +1

CH m ∪ N -1 CH m . Using the P U (m, 1) invariance, it is easy to see that this is the whole N CH m . The nonnegativity statement is automatic and the fact that µ g , up to P U (m, 1), depends only on g is like in the odd case.

We are left to justify the rigidity part. If the mass µ g vanishes, we know by construction that every element φ of K c CH m gives rise to an element ψ of K c g that is asymptotic to φ. In particular, for any element

of N g that is asymptotic to β. As a consequence, N g has maximal dimension : (M, g) is complex hyperbolic.

Appendix : an example.

To build a non-trivial example of asymptotically complex hyperbolic Kähler manifold, we use the symplectic point of view of [HS]. We work on C m , m ≥ 2 and let ρ = e t be the Euclidean distance to the origin. Then dd c t = ω F S is the pull-back to C m \ {0} of the Fubini-Study form on CP m-1 . Every U (m)-invariant Kähler structure on C m \ {0} admits a Kähler form reading ω = ψ ′ (t)dt ∧ d c t + ψ(t)ω F S , where ψ is simply the derivative of a radial Kähler potential with respect to the variable t. The function ψ and ψ ′ must of course be positive. In fact ψ can be seen as a moment