The DARIAH ERIC: Redefining Research Infrastructure for the Arts and Humanities in the Digital Age by Jennifer Edmond (Trinity College Dublin), Frank Fischer (National Research University Higher School of Economics, Moscow), Michael Mertens (DARIAH EU) and Laurent Romary (Inria) As it begins its second decade of development, the Digital Research Infrastructure for the Arts and Humanities (DARIAH) continues to forge an innovative approach to improving support for and the vibrancy of humanities research in Europe.

When we think of infrastructure, we often fall back on canonical images of things like roads, bridges and buildings. In many disciplines, these still resonate with the needs of the researcher community, where a supercollider or a research vessel may indeed be at the core of what is required to advance our state of knowledge.

The arts and humanities are different. As a collection of approaches to knowledge, the methods deployed stem from a shared respect for complex source material emerging not from the experimental design of the scientist, but from the experiences, cultures and creative impulses of human beings. Providing an enhanced, shared, baseline access to key methods, sources, tools and services is therefore a great challenge indeed.

The Digital Research Infrastructure for the Arts and Humanities (DARIAH) was first conceptualised in late 2005 as a response to how this very different set of requirements was being addressed in the fast-moving environment of digitallyenhanced research. The infrastructure was later officially founded as a European Research Infrastructure Consortium (or ERIC) based in France, but with 17 national members contributing funds and in-kind contributions from their local digital humanities research communities. The knowledge base of the resulting network is further enhanced by contributions from funded research projects in which DARIAH is a partner, as well as the contributions of working groups, assembled by network members on a voluntary basis to address key gaps in infrastructural provision or key emerging challenges for the research communities. This rich tapestry of contributions creates a form of infrastructure based on knowledge production and exchange, rather than on concrete shared facilities or datasets. As such, the challenge for DARIAH as it enters its second decade of development is to capitalise on the human infrastructure it has built to create a fully aligned system of coordinated contribution and access provision to the good practices emerging from the network [START_REF] Blanke | Crowds for Clouds: Recent Trends in Humanities Research Infrastructures[END_REF]. In order to do this, we are focussing for the next three years on four key areas of development that will enhance our ability to deliver low-friction, high value interactions for our partners.

The first area of focus is to improve our external communications. Ensuring that our basic information is available in an easily legible form for all current and potential new members of our network is a sine qua non for ensuring that the infrastructure can function. This programme of activities will cover the gamut of communications instruments, from a newly redesigned website to the appointment of specific individual ambassadors to reach key target regions and communities; from a more strategic approach to attending events to a mapping of key organisation-toorganisation relationships within our community and beyond. We will also clarify what we are able to offer as services to our community, from support for grant capture to hosting of orphan research projects. By focussing on our core messages in this way, we hope also to be able to communicate and build consensus around an even clearer message of what DARIAH is and does, and how it operates as an infrastructure in conditions that require a very different approach.

The second plank in DARIAH's development plan is to push forward its vision for a virtual marketplace, making visible and accessible the many tools, services, datasets and expertise bases that our network has opened up for use by others. This may sound like an easy task to achieve, but the prerequisite understanding of what these assets would be valued for and by whom is actually quite challenging to develop. Ensuring that we provide an optimised platform for targeted and serendipitous discovery of resources, as well as their easy reuse, will be a major achievement of DARIAH by 2020.

Our third area of focus is on teaching and learning. Too much focus in the digital humanities is on either training via formal degree programmes or through individual learning via generic platforms like Code Academy or Software Carpentry. The research infrastructure provides a unique environment and set of opportunities for different kinds of learning, aligned to support the individual and institutional modes, but also to provide unique opportunities for experiences of professional acculturation in applied contexts [START_REF] Rockwell | Acculturation and the Digital Humanities Community[END_REF]. Already in this area we have active services under the banners of dariahTeach, a Moodle-based, ECTS-linked set of modules, and through our infrastructure cluster project PARTHENOS, which involves the CLARIN ERIC and projects such as IPERION, CHARISMA, ARIADNE, EHRI and CENDARI as well, and where the modules are more targeted at self-learners and as "train-the-trainers" resources. We will now build on these platforms and momentum.

Finally, we view the development of our foresight and policy leadership capacity as a key asset, not only for our current cohort of active digital humanists, but also for the "long tail" of the research communities. These researchers, who may not realise how important the digital is becoming for how they conduct and communicate their research, are a key community for our future growth, and ensuring that we represent their needs at a high level is of central importance to us as we consolidate our position. Our work to date in this area has been linked to initiatives such as DARIAH's contributions to the European Commission's Open Science Policy Platform, and also through our championing of a "Data Reuse Charter" between researchers and cultural heritage institutions, able to promote data sharing and fluidity [START_REF] Romary | Data fluidity in DARIAHpushing the agenda forward[END_REF].

On the basis of these interventions, DARIAH is poised to move into its second decade with a reputation as a leader for the arts and humanities, as well as an innovator in research infrastructure.


Edward S Beetham
email: e.beetham@auckland.ac.nz

Paul S Kench

Stéphane Popinet

Future reef growth can mitigate physical impacts of sea-level rise on atoll islands

Keywords: Atoll, Reef island, Sea-level rise, Wave transformation, Wave overtopping, Infragravity wave

come   L'archive ouverte pluridisciplinaire

Introduction

Future habitation on low-lying atoll islands is at risk from an increase in flooding and wave inundation associated with sea-level rise (SLR) and anthropogenic climate change [START_REF] Storlazzi | Many Atolls May be Uninhabitable Within Decades Due to Climate Change[END_REF]. As geological structures, coral islands are accumulations of reef-derived biogenic sediment deposited on atoll and platform reefs, with elevations rarely exceeding 2-3 m above mean sea-level (MSL) [START_REF] Woodroffe | Reef-island topography and the vulnerability of atolls to sea-level rise[END_REF][START_REF] Mclean | Destruction or persistence of coral atoll islands in the face of 20th and 21st century sea-level rise?[END_REF]. Rising sealevels are expected to result in larger waves at reef island shorelines, increasing the frequency and magnitude of wave generated flooding events [START_REF] Sheppard | Coral mortality increases wave energy reaching shores protected by reef flats: Examples from the Seychelles[END_REF][START_REF] Quataert | The influence of coral reefs and climate change on wave-driven flooding of tropical coastlines[END_REF]. Wave inundation is a major concern for reef island communities, with negative impacts including physical damage to homes and critical infrastructure, destruction of food resources and contamination of freshwater reservoirs [START_REF] Connell | Losing ground? Tuvalu, the greenhouse effect and the garbage can[END_REF][START_REF] Lazrus | Sea Change: Island Communities and Climate Change[END_REF][START_REF] Lin | Interpreting and analyzing King Tide in Tuvalu[END_REF][START_REF] Nurse | Small islands, Clim. Chang. 2014 Impacts, Adapt. Vulnerability[END_REF]. Under contemporary conditions, wave overtopping on reef islands typically requires a large wave event coincident with highest tidal levels, a combination of factors that occurs every 2-3 years [START_REF] Merrifield | Observations and estimates of wavedriven water level extremes at the Marshall Islands[END_REF]. However, more frequent wave overtopping associated with SLR will limit recovery time between events and may strain the coping capacity of island communities to a point that seriously compromises permanent habitation [START_REF] Barnett | Climate Dangers and Atoll Countries[END_REF][START_REF] Patel | Climate science: A sinking feeling[END_REF].

Recent analysis of wave interaction with atoll reefs under SLR scenarios suggest that the tipping-point when islands may become uninhabitable will occur in a few decades, when runup under modal wave conditions exceeds the berm elevation [START_REF] Storlazzi | Many Atolls May be Uninhabitable Within Decades Due to Climate Change[END_REF].

However, previous investigations on the physical impact of SLR on reef islands omit two important negative feedback mechanisms that may offset the imminent threat of SLR for island communities. First, existing studies do not explicitly resolve the contribution of secondary wave motions (infragravity waves and wave setup) to island flooding or examine how these processes will respond to higher sea-level. Second, there has been no attempt to This article is protected by copyright. All rights reserved.

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quantify the effectiveness of future vertical reef accretion on mitigating the physical impact of SLR on reef islands.

Wave overtopping is influenced by three key wave processes, incident frequency sea-swell (SS) waves (f > 0.04 Hz), long period infragravity (IG) waves (f ≤ 0.04 Hz) and water level setup shoreward of the breakpoint [START_REF] Merrifield | Observations and estimates of wavedriven water level extremes at the Marshall Islands[END_REF]. Coral reefs are known to be highly effective at dissipating SS waves, reducing incident energy by as much as 97% before island shorelines [START_REF] Ferrario | The effectiveness of coral reefs for coastal hazard risk reduction and adaptation[END_REF]. However, SS wave dissipation facilitates a transfer of energy to longer period (IG) wave motions which are consistently recognized as the dominant process influencing wave runup [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF], overtopping [START_REF] Cheriton | Observations of wave transformation over a fringing coral reef and the importance of low-frequency waves and offshore water levels to runup, overwash, and coastal flooding[END_REF] and destructive flooding [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF][START_REF] Shimozono | Combined infragravity wave and sea-swell runup over fringing reefs by super typhoon Haiyan[END_REF] on reef shorelines. The presence of IG waves on coral reefs is associated with two key forming mechanisms: (1) the release of group-bound long waves when incident waves break [START_REF] Longuet-Higgins | Radiation stress and mass transport in gravity waves, with application to "surf beats[END_REF][START_REF] Masselink | Group bound long waves as a source of infragravity energy in the surf zone[END_REF] and (2) the propagation of dynamic setup arising from spatial variability in break-point location [START_REF] Symonds | Two-dimensional surf beat: Long wave generation by a time-varying breakpoint[END_REF]. Both forcing mechanisms are associated with irregular incident wave groups and result in freepropagating long period waves on the reef flat [START_REF] Péquignet | Energy transfer between wind waves and low-frequency oscillations on a fringing reef, Ipan, Guam[END_REF]. The development of IG waves with enough energy to overtop island shorelines requires a substantial reduction in SS energy through breaking and friction, which implies that IG energy may become less important at higher sea-levels if SS dissipation decreases [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF]. Accurate predictions of wave driven flooding on reef coastlines must account for this nonlinear behavior between SS and IG frequency waves [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF].

To date, studies that have examined the impact of SLR on wave transformation across coral reefs have assumed a static reef platform surface. However, many coral reefs are currently sea-level limited and future SLR will open accommodation space for vertical reef accretion [START_REF] Perry | Estimating rates of biologically driven coral reef framework production and erosion: a new census-based carbonate budget methodology and applications to the reefs of Bonaire[END_REF][START_REF] Woodroffe | Coral reefs and sea-level change[END_REF]. Reef accretion represents a feedback to This article is protected by copyright. All rights reserved.

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existing interpretations of increased wave impacts and overtopping on islands that may provide a self-regulating mechanism to mitigate the physical impact of SLR on reef islands.

To date such feedbacks have not been examined. The potential for future reef growth is the subject of continued speculation [START_REF] Van Woesik | Keep up or drown: adjustment of western Pacific coral reefs to sea-level rise in the 21st century[END_REF]. Conventional wisdom suggests reef growth is likely to be compromised in many reef regions as a consequence of a global decline in coral cover, increased sea surface temperatures, ocean acidification and anthropogenic stresses [START_REF] Jackson | Historical Overfishing and the Recent Collapse of Coastal Ecosystems[END_REF][START_REF] Hughes | Climate Change, Human Impacts, and the Resilience of Coral Reefs[END_REF][START_REF] Hughes | Rising to the challenge of sustaining coral reef resilience[END_REF]Hoegh-Guldberg et al., 200fig. 7fig;[START_REF] Hoegh-Guldberg | Coral reef sustainability through adaptation: glimmer of hope or persistent mirage?[END_REF][START_REF] Aronson | Physical and Biological Drivers of Coral-Reef Dynamics[END_REF]. However, recent studies have documented coral re-colonization across previously emergent reef flats due to SLR [START_REF] Brown | Increased sea level promotes coral cover on shallow reef flats in the Andaman Sea, eastern Indian Ocean[END_REF][START_REF] Scopélitis | Coral colonisation of a shallow reef flat in response to rising sea level: quantification from 35 years of remote sensing data at Heron Island, Australia[END_REF] and land subsidence [START_REF] Saunders | Tectonic subsidence provides insight into possible coral reef futures under rapid sea-level rise[END_REF],

providing evidence that locally some reefs may have capacity to vertically accrete and keep pace with future sea-levels. Indeed, recent field and modeling assessments show that sealevel constrained reefs in the western tropical Pacific are currently accreting in response to measured SLR and will continue to keep-up with future SLR at rates up to 6.1 mm/yr, even when negative effects of increasing sea-surface temperature are represented [START_REF] Van Woesik | Keep up or drown: adjustment of western Pacific coral reefs to sea-level rise in the 21st century[END_REF]. Collectively, such studies suggest future reef growth responses are likely to vary between reef localities dependent on the antecedent condition of reefs and the severity of future pressures [START_REF] Kench | Coral Reefs, in Geomorphology and global environmental change[END_REF][START_REF] Hamylton | Simulating reef response to sea-level rise at Lizard Island: A geospatial approach[END_REF][START_REF] Perry | Remote coral reefs can sustain high growth potential and may match future sea-level trends[END_REF]. In areas where reef accretion does occur, the effect on modifying wave processes and mitigating hazard risks to reef islands remain poorly resolved.

A third limitation of existing research on the physical impacts of SLR is the inability of wave transformation based studies to appropriately account for ongoing adjustments to island morphology as shoreline wave energy increases. Accounting for the morphodynamic impact of SLR on atoll islands is beyond the scope of this study but will be the focus of future research. The focus of this study is to resolve how SLR will modify the nonlinear wave Accepted Article transformation processes that cause flooding and provide the boundary condition for island morphology change, accounting for static and accretionary reef responses. Recent advances in numerical modeling have developed phase-resolving techniques capable of accurately resolving water level motions associated with SS waves, IG waves, wave setup, runup and overtopping on coral reefs, providing an unprecedented ability to represent nonlinear wave transformation on morphologically complex reef environments [START_REF] Roeber | Boussinesq-type model for energetic breaking waves in fringing reef environments[END_REF][START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF][START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF]. This study is focused on Fatato Island, Funafuti Atoll, Tuvalu where previous research was done to evaluate wave transformation outputs from a fully nonlinear Boussinesq model against field measurements [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF]. In this study, the same model was utilized to simulate wave transformation, runup and overtopping on Fatato with incremental SLR and variable reef response morphologies.

Results are used to explore how long islands on the eastern rim of Funafuti will remain inhabitable before frequent inundation may compromise human occupation. This approach does not account for adjustments to island morphology and therefore assumes that ongoing habitation requires islands to maintain their current morphology.

Field setting

Fatato is located on the south-eastern rim of Funafuti Atoll and is comprised of sand and gravel size material, with a morphology representative of many high energy 'motu' islands throughout the Pacific Ocean, including neighboring island Fogafale, where approximately 6,000 people live (Fig. 1). Funafuti is located inside a SLR hotspot, where MSL increased by 0.3 m between 1950 and 2009 (5.1 mm yr -1 ) [START_REF] Becker | Sea level variations at tropical Pacific islands since 1950[END_REF]. The atoll has been a focus for numerous studies examining climate change vulnerability [START_REF] Farbotko | The first climate refugees? Contesting global narratives of climate change in Tuvalu[END_REF][START_REF] Lazrus | Sea Change: Island Communities and Climate Change[END_REF][START_REF] Kench | Coral islands defy sea-level rise over the past century: Records from a central Pacific atoll[END_REF][START_REF] Mccubbin | Where does climate fit? Vulnerability to climate change in the context of multiple stressors in Funafuti, Tuvalu[END_REF] but existing research has not considered how increasing sea-level (SL) will change the physical processes that contribute to wave driven flooding. The ocean facing reef flat adjacent Fatato Island is approximately This article is protected by copyright. All rights reserved.

Accepted Article

100 m wide and the offshore reef slope is approximately 23.5°. Conglomerate platform and cemented rubble protrude seaward from the island base, with the beach toe elevated 0.39 m above MSL. The model transect from Fatato Island that was used in these experiments is representative of the typical topography of many islands on the eastern rim of Funafuti Atoll.

The ocean shoreline has a steep gravel beach face (12° gradient) with a berm elevated 3.5 m above MSL (4.3 m above the central reef flat). The beach face and island are comprised entirely of calcium carbonate sediment derived from physical breakdown of the reef framework. Mean grain size on the beach face is 73 mm, with most clasts within the pebble or cobble size fractions, although boulder size clasts are scattered throughout the island and small sand pocks do exist on the beach face. Fatato Island is 80 -90 m wide, with a basin in the center and a ridge on the lagoon shoreline (Fig. 1d).

The eastern reef flat on Funafuti Atoll is currently intertidal, with the outer reef flat elevated approximately 0.3 m above spring low tide levels. The potential for coral colonization and therefore vertical accretion on the reef flat will increase when SLR results in permanent submergence of the contemporary reef. Detailed reef drilling records show that the reef flat at Funafuti accreted by 26.4 m during the Holocene, at a mean rate of 5.0 mm/yr [START_REF] Ohde | The chronology of Funafuti Atoll: revisiting an old friend[END_REF]. Two hypothetical reef growth morphologies were used in this research, based on an understanding of morphodynamic zonation of wave exposed reefs [START_REF] Hopley | Climate Change: impact of sea level rise on reef flat zonation and productivity[END_REF][START_REF] Hamylton | Simulating reef response to sea-level rise at Lizard Island: A geospatial approach[END_REF]. The first morphology concentrates reef accretion at the outer reef flat, representing a concentration of coral near the reef edge or the development of a prominent coralline algae and rubble ridge formation [START_REF] Guilcher | Coral reef geomorphology[END_REF], with limited coral cover on the inner reef (Fig. 2). The second morphology applies a near-uniform level of reef accretion across the reef flat, representing coral colonization or sedimentary infill on the inner reef flat.

Maximum elevation for both morphologies is at the same elevation as the SLR adjusted

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spring low tide level (Fig. 2). Both reef accretion morphologies assume no change to the volume of island sediment.

Methods

Reef and island topography

A laser level total station was used to measure reef and island topography on 10 across reef transects including the profile used for numerical model analysis. The profiles were combined with RTK-GPS survey points from the reef flat to create a terrain model of the reef flat and shoreline. Shallow water topography data were combined with satellite imagery and single beam echo-sounding data [START_REF] Hoeke | Downscaling wave climate at a Funafuti, Tuvalu, Report[END_REF] to create a bathymetry map of the atoll reef flat near Fatato Island (Fig. 1c). All references to topographic elevation are relative to MSL = 0.

The 1D model transect used for wave simulations is a composite of the total station measured island and reef flat profile and the reef slope bathymetry at the instrument transect location (Fig. 1c). The total station profile provided coverage of Fatato Island until the outer reef flat with a seaward limit 1.2 m below MSL, 90 m from the beach toe (Fig. 1d). The transect was extended seaward by overlapping and stitching a profile sliced from the atoll bathymetry that provided 10 m resolution coverage of the outer reef flat, the reef edge and reef slope to the 100 m depth contour (Fig. 1c). Total station and bathymetry data were both referenced to x = 0 m at the toe of beach and interpolated to 1 m resolution for numerical model simulations.

The model domain was 1,024 m long, with imported waves propagating for 700 m over a uniform depth of 100 m before reaching the atoll reef profile as indicated by measurements of reef bathymetry and island topography data [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF].

Numerical model

The fully nonlinear Green-Naghdi solver model described in [START_REF] Popinet | A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations[END_REF] was used to simulate wave interaction with the reef and island. The model was previously evaluated

Accepted Article

against field data from Fatato, achieving model skill > 0.96 for predictions of Hss (significant wave height for the SS band), Hig (significant wave height for the IG band) and setup elevation at the shoreline, across 500 different incident wave and tide combinations [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF]. The numerical model has also been validated against a series of benchmark test scenarios for runup and overwash [START_REF] Popinet | A solver for the Green-Naghdi equations[END_REF][START_REF] Popinet | A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations[END_REF][START_REF] Beetham | Field and numerical investigations of wave transformation and inundation on atoll islands[END_REF]. These scenarios include, runup on a planar beach, runup on a conical island, waves overtopping a sea wall, runup on a coral reef shoreline, overtopping a reef crest and overtopping on a complex shelf with a conical island. The phase-resolving model simulates dispersion using the Green-Naghdi equations and locally switches to a nonlinear shallow water system to represent wave breaking when the free-surface slope exceeds 1 (45°) [Bonneton et al., 2011a;[START_REF] Popinet | A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations[END_REF].

A benefit of this fully nonlinear phase-resolving approach is that no specific parameterization is required to represent surf-zone dynamics associated with IG waves and setup because the model appropriately accounts for momentum balances created by an irregular incident wave field [START_REF] Bonneton | Large-Scale Vorticity Generation Due to Dissipating Waves in the Surf Zone[END_REF][Bonneton et al., , 2011b]]. Similar models have been shown to accurately represent the formation and behavior of IG waves gentle sloping beaches [START_REF] Tissier | Fully nonlinear Boussinesq-type modelling of infragravity wave transformation over a lowsloping beach[END_REF] and on coral reefs [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF][START_REF] Shimozono | Combined infragravity wave and sea-swell runup over fringing reefs by super typhoon Haiyan[END_REF].

Model experiments

Boundary conditions

Model simulations presented here encompass the contemporary spring tidal range (SL = -1.0 m to SL = 1.0 m) and represent a range of SLR magnitudes from 0.05 m to 1.5 m above present spring high (SL = 1.05 m to SL = 2.5 m), relative to MSL = 0. Three non-extreme wave conditions were used, based on analysis of 34 year wave hind-cast data offshore the eastern rim of Funafuti Atoll [START_REF] Durrant | A global wave hindcast focussed on the Central and South Pacific[END_REF][START_REF] Hoeke | Downscaling wave climate at a Funafuti, Tuvalu, Report[END_REF]. Wave conditions include, a) mean significant wave height and period (Hs = 1.3 m; Ts = 10 s), and the average annual-maximum wave heights associated with a) local storm activity (Hs = 2.6 m; Ts = 10 s)

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and b) long period swell (Hs = 2.1 m; Ts = 16 s). Measured wave field data offshore Fatato from [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF] were used as a boundary condition to ensure accurate representation of infragravity wave dynamics and setup magnitude. The surface water level time-series representing each wave condition was imported into the offshore model boundary (h = 100 m) to initiate each simulation (Fig. 3). Mean waves were characterized by low energy across a range of frequencies with distinct energy peaks at 0.079 and 0.1 Hz (Fig. 3a).

Swell waves had the largest peak in spectral density, centered at 0.06 Hz (Fig. 3b). Storm waves had a lower peak spectral density than swell waves but the wide distribution of high energy centered around 0.95 Hz resulted in the largest gross energy of all wave conditions used in this analysis (Fig. 3c). More information on model calibration and performance for simulating waves on Fatato is available in [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF].

Simulations involving sea-level rise with a static reef response

Assuming no morphological response from the reef or island, across reef wave transformation for each of the three boundary wave conditions was simulated using 71 different sea-levels (213 separate simulations). Sea-level (SL) was increased at 0.05 m increments, from SL = -1 (mean spring low) to SL = 2.5 (mean spring high + 1.5 m).

Simulations involving sea-level rise with a keep-up reef response

The same wave conditions were used to simulate how wave processes will change if the reef flat exhibits a vertical growth response to SLR. Simulations of high, mean, and low tide stages were undertaken for SLR = 0, 0.5, 1 and 1.5 m respectively, and repeated using the two previously described reef response morphologies.

Model outputs and analysis

Each simulation ran for the 2048 second (~35 minute) duration of the input wave field timeseries. This time period ensures that multiple wave groups were represented in each simulation. Only model outputs after 512 s were used to calculate wave and overtopping This article is protected by copyright. All rights reserved.

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statistics to ensure the wave field had fully developed across the reef flat. Time-series data were extracted at 10 Hz at 5 m intervals across the reef flat, with outputs at the shoreline sensor (0.38 m below MSL) and the berm crest (MSL + 3.5 m) the main focus of this analysis (Fig. 1d). A 0.04 Hz band pass filter was used to separate high and low frequency waves and the zero down-crossing method was used to quantify significant wave height for sea-swell waves (Hss) and infragravity waves (Hig). Power spectral density (PSD) calculations from the raw time-series were used to quantify total spectral density within the wave field (0.0037-2 Hz), total spectral density within the SS wave band (0.04-2 Hz) and total spectral density within the IG wave band (0.0037-0.04 Hz). Wave overtopping events were identified and OW rates were calculated using time-series data extracted from the berm crest (Fig. 1d). OW rate is presented as the mean volume of water passing over the berm during the final 1,536 seconds (~25 minutes) of each simulation and is measured in cubic meters per second, assuming the model represents a 1 m wide section of the island. Significant levels of overwash can result in the central island basin filling with water, however this does not interfere with calculated OW rates on the ocean berm because drainage occurs on the lower elevation leeward island ridge.

Results

Sea-level rise with a static reef response

Shoreline wave processes

Model simulations with a static reef morphology indicate that a fundamental shift in the shoreline process regime will occur on reefs that exhibit no morphological adjustment to future SLR (Fig. 4). First, model results demonstrate a near linear increase in SS wave height (Hss) at the shoreline as sea-level increases from spring low tide (SL = -1.0 m) to spring high tide (SL = 1 m), with continued increase in Hss towards SL = 2.5 m, encompassing SLR scenarios between 0.05 and 1.5 m above the contemporary tidal range (Fig. 4a-c). The This article is protected by copyright. All rights reserved.

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positive relationship between wave height and reef depth is accompanied by a near exponential decrease in setup elevation at the shoreline, as less energy is dissipated through wave breaking and friction (Fig. 4a-c). The relationship between shoreline IG wave height (Hig) and SLR is more complex (Fig. 4d-f). Under each wave condition Hig initially increased as reef water depth increased from SL = -1 m, but maximum Hig was observed within the current tidal range, at SL = 0 m, 0.35 m and 0.65 m for mean, swell and storm waves respectively (Fig. 4g-i). Beyond these maximum values, Hig decreased as sea-level increased towards current spring high tide and beyond present levels (Fig. 4). Model outputs show that an inert reef response to rising sea-level will shift the wave environment at island shorelines from one currently dominated by IG energy to a regime where incident frequency SS waves are the primary source of hydrodynamic interaction with island shorelines (Fig. 4d-f). Indeed, model outputs show a marked increase in net shoreline wave energy once SS waves accounted for more than 50% of shoreline energy (Fig. 4g-i). This threshold sea-level threshold for SS waves becoming the dominant source of energy is reached around high tide under mean wave conditions and at low levels of SLR for swell and storm waves respectively (Fig. 4).

Wave overtopping

Importantly, model results show that wave overtopping of the island shoreline does not occur within the contemporary tidal range under annual-maximum storm or swell conditions (Fig. 4j-l). Larger wave events associated with decadal to centennial scale return periods are capable of overtopping islands on Funafuti Atoll (e.g. Cyclone Bebe in 1972), but this study focused on modal wave conditions that if capable of overtopping islands may significantly compromise permanent habitation [START_REF] Storlazzi | Many Atolls May be Uninhabitable Within Decades Due to Climate Change[END_REF]. Island overtopping under the annual storm and swell conditions used in this study initially occur at spring high tide with SLR = 0.3 m and 0.35 m, respectively (SL = 1.3 m and 1.35 m). A threshold sea-level of 1.9

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m above MSL (spring high tide plus 0.9 m of SLR) is required for island overtopping under mean wave conditions. Once the SL threshold was reached for each wave condition, overwash (OW) volume increased exponentially, and overtopping moved from a single wave surge (OW < 0.003 m 3 /s) toward multiple wave groups (OW > 0.05 m 3 /s) breaching the berm (Fig. 4j-l). Unlike contemporary observations of overtopping, model outputs identify SS waves will provide the primary source of overwash flow with SLR, under non-extreme wave.

Notably, swash motions generated by IG waves are capable of surging landward with considerable momentum and significantly exceed the reach of SS waves [START_REF] Guza | Swash oscillations on a natural beach[END_REF]. Therefore, SS wave driven overtopping on reef islands at higher sea-levels under nonextreme waves will be fundamentally different to the IG dominated and destructive overtopping events observed during extreme wave conditions at contemporary sea-level [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF][START_REF] Cheriton | Observations of wave transformation over a fringing coral reef and the importance of low-frequency waves and offshore water levels to runup, overwash, and coastal flooding[END_REF].

Effectiveness of vertical reef accretion at protecting reef islands

Shoreline wave processes

Wave processes at the shoreline associated with SLR and vertical reef adjustment are not identical to contemporary conditions because the reef flat is currently elevated 0.3 m above spring low tide compared to the adjusted reefs being level with spring low tide. However, model simulations representing vertical reef accretion do indicate that much of the projected increase in shoreline wave energy can be mitigated through natural reef morphology feedbacks (Fig. 5). SS wave height at high tide remained within 10% and 5% of contemporary high tide values for simulations with outer reef accretion and full reef accretion, respectively. In comparison, SS wave height at high tide increased by 20% for every 0.5 m of SLR when the reef remained static (Fig. 5). The contemporary reef is effective at dissipating 54.6% of incident wave height at high tide, when averaged across the three wave conditions. Reef accretion maintains this same magnitude of dissipation. In comparison, This article is protected by copyright. All rights reserved.

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a static reef response is associated with 30% and 40% less wave dissipation at SLR = 1.0 m and 1.5 m, respectively. Averaged across all simulated wave conditions and tidal stages (low, mean, high), vertical accretion of the outer reef flat was effective at offsetting 15-57% of the increase in SS wave height associated with a static response to SLR, depending on the SLR magnitude (Fig. 5). Comparatively, full reef flat accretion was capable of dampening the SLR associated increase in SS wave height by 33-79%.

Both accretionary morphologies resulted in larger IG waves at the shoreline under each SLR scenario when compared to SLR and a static reef morphology (Fig. 5; Table S1). IG wave heights associated with both reef accretion morphologies remained within 10% of contemporary magnitudes at high tide, on average, with a slight decrease in height due to the contemporary reef being intertidal (Fig. 5d-f). In contrast, SLR with a static reef morphology resulted in a dramatic reduction in shoreline IG wave height, with a 10% decrease in height associated with every 0.5 m increase in sea level. Model outputs also show that SLR will result in larger IG wave heights impacting atoll islands at low tide and mid tide, regardless of reef response morphology or wave condition (Table S1).

An inert reef response to SLR resulted in setup decreasing at mid and high tide, with an increase in setup only observed at low tide under conditions when the shoreline was previously dry. Reef accretion resulted in similar setup magnitudes to contemporary conditions with setup 54-60% larger than the static reef morphology for outer and full reef accretion, respectively (Fig. 5).

Wave overtopping

Model results show that a vertical growth response to SLR is effective at decreasing overwash volume, given the assumption that island morphology remains static in all simulations (Fig. 5). However, model outputs also show that reef accretion has limited potential to prevent overtopping or offset the SLR threshold when overwash occurs for a This article is protected by copyright. All rights reserved.

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particular wave condition. For storm waves, outer reef accretion reduced OW volume by 46% on average, increasing to 65% reduction with full reef flat development (Fig. 5; Table S2).

Reef growth was less effective at preventing overtopping under swell conditions, with OW volume reduced by 27% and 57% for outer and full reef growth morphologies, respectively (Table S2). Reef growth was most effective at decreasing OW under mean wave conditions, with an average reduction of 78% and 93%, respectively. Overall, an accretionary response to SLR will substantially reduce the magnitude of OW when compared to a static response.

However, model results emphasize that future reef growth has minimal influence on preventing the occurrence of wave overtopping with SLR and highlight that reef accretion has limited potential to offset the SL threshold when overtopping first occurs (Fig. 6).

Discussion

Previous studies on the impacts of SLR on reef island flooding are based on the assumption that reef morphology will remain static and exhibits no structural adjustment to SLR [START_REF] Storlazzi | Many Atolls May be Uninhabitable Within Decades Due to Climate Change[END_REF]. Despite the multiple and compounding stressors that will likely limit future reef growth [START_REF] Anthony | Ocean acidification and warming will lower coral reef resilience[END_REF][START_REF] Hoegh-Guldberg | Coral Reef Ecosystems under Climate Change and Ocean Acidification[END_REF][START_REF] Hughes | Coral reefs in the Anthropocene[END_REF], recent field assessments have measured vertical reef accretion in response to relative or absolute SLR [START_REF] Brown | Increased sea level promotes coral cover on shallow reef flats in the Andaman Sea, eastern Indian Ocean[END_REF][START_REF] Scopélitis | Coral colonisation of a shallow reef flat in response to rising sea level: quantification from 35 years of remote sensing data at Heron Island, Australia[END_REF][START_REF] Saunders | Tectonic subsidence provides insight into possible coral reef futures under rapid sea-level rise[END_REF][START_REF] Van Woesik | Keep up or drown: adjustment of western Pacific coral reefs to sea-level rise in the 21st century[END_REF]. Model based projections from field data in the western Pacific [START_REF] Van Woesik | Keep up or drown: adjustment of western Pacific coral reefs to sea-level rise in the 21st century[END_REF] suggest that some reefs can potentially keep-up with future SLR forced by RPC4.5 (6.1 mm/yr) but will likely lag behind RPC6.0 scenarios (7.4 mm/yr), with limited potential for a keep-up reef response under RPC8.5 (11.2 mm/yr). Previous hydrodynamic studies have also relied on phase-averaged models that do not adequately resolve infragravity wave processes, setup, runup and overwash motions [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF]. These secondary surf-zone processes provide a primary contribution to runup [START_REF] Beetham | Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu[END_REF] and inundation [START_REF] Cheriton | Observations of wave transformation over a fringing coral reef and the importance of low-frequency waves and offshore water levels to runup, overwash, and coastal flooding[END_REF] on atoll islands. Results presented here offer unique and This article is protected by copyright. All rights reserved.

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comprehensive insight on how SLR will change the full spectrum of wave processes that contribute to island flooding and provide the first assessment regarding how effective future reef growth is at mitigating adverse effects of SLR. Consistent with previous studies, SLR with no reef-adjustment will increase reef depth and allow larger waves within the SS frequency band to impact reef island shorelines. However, model results from Funafuti also show that as sea-level increases and more energy is retained within the SS frequency band, less energy is transferred into the IG wave band. Model results identify that shoreline wave energy transitions from being IG dominant to being SS dominant when reef submergence reaches a critical threshold of Ho/hr < 1.3 (where hr is depth at the reef edge). Our findings contrast previous studies that predict an increase in IG wave height with SLR because of the associated decrease in bottom friction [START_REF] Quataert | The influence of coral reefs and climate change on wave-driven flooding of tropical coastlines[END_REF]. These contrasting results may be associated with differing reef widths or the way different models represent nonlinear interactions between SS and IG waves. Similar to IG waves, setup water level on coral reefs will decrease with SLR, as presented here and previously predicted using empirical data [START_REF] Becker | Water level effects on breaking wave setup for Pacific Island fringing reefs[END_REF].

A significant advantage of fully nonlinear phase-resolving models is that the free-surface boundary condition can directly simulate runup and overtopping flow. Previous wave modeling based studies of flooding on atoll islands have relied on modified runup equations based on shoreline wave height to predict flooding magnitudes [START_REF] Quataert | The influence of coral reefs and climate change on wave-driven flooding of tropical coastlines[END_REF][START_REF] Storlazzi | Many Atolls May be Uninhabitable Within Decades Due to Climate Change[END_REF]. Another limitation of previous research that remains unresolved in this study is the unknown morphological response of atoll islands to SLR. Previous research on the potential for islands to adjust their morphology in response to SLR are limited in number and scope, but there is a hypothesis that some islands will accrete vertically and migrate lagoon-ward as overwash processes re-work new and existing island sediment [START_REF] Cowell | The morphological response of atoll islands to sealevel rise. Part 1: modifications to the shoreface translation model[END_REF][START_REF] Kench | The morphological response of atoll islands to sealevel rise. Part 2: application of the modified shoreface translation model (STM)[END_REF]. All results in this study assume no island adjustment This article is protected by copyright. All rights reserved.

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to SLR and therefore overtopping values do not account for how overtopping events may result in changes to the shoreline position and elevation. To justify this omission we argue that the dramatic change in island morphology required to sustain island elevation above rising sea levels will itself compromise ongoing habitation on atoll islands.

With a static reef and island response to SLR, overtopping under annual conditions will initially occur on the eastern rim of Funafuti at spring high tide with 0.3 m of SLR.

Overtopping under annual-maximum wave events will occur within four decades if the current rate of SLR (5.1 mm/yr) prevails or within two decades if emissions continue to increase under IPCC business-as-usual projections (Table 1). Model outputs for storm and swell waves show that overwash between SLR = 0.3 and 0.85 m is associated with 2 or 3 waves overtopping the berm in a 35 minute burst, resulting in low volumes of flooding that are potentially manageable for island communities. Under these annual-maximum conditions, spring high tide overwash volume increases substantially when SLR exceeds 0.9 m, as multiple wave groups are able to breach the island berm, causing a significant increase in flood volume. The threshold for wave generated flooding to occur under mean wave condition at spring high tide was also SLR = 0.9 m (Fig. 4), highlighting this as a conservative tipping-point for the frequency and magnitude of wave generated flooding to overwhelm recovery potential. Mild overtopping by mean waves and serious overtopping from annual-maximum events (SLR = 0.9 m) will occur by 2090 under business-as-usual SLR projections, and will seriously strain the resilience of inhabited reef islands. The change in hydrodynamic boundary conditions at island shorelines will likely result in significant shoreline change on natural atoll islands over the next 70 years, while significant flooding will occur on islands with urbanized shorelines. Based on IPCC rates of SLR under lower emission scenarios, the onset of serious wave driven flooding can be significantly delayed if global action is taken to reduce emissions (Table 1). Under modal wave conditions, This article is protected by copyright. All rights reserved.

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overtopping with SLR will be driven by incident frequency SS waves and not long period infragravity waves, which are consistently identified as the primary driver of overtopping and destructive flooding on reef shorelines [START_REF] Roeber | Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan[END_REF][START_REF] Cheriton | Observations of wave transformation over a fringing coral reef and the importance of low-frequency waves and offshore water levels to runup, overwash, and coastal flooding[END_REF].

Model outputs show that future reef accretion will effectively preserve the contemporary tidal modulation of wave processes at the shoreline and will significantly mitigate wave flooding magnitudes. The inability of reef accretion to prevent overtopping is partly attributed to IG waves and setup compensating for the reduction in SS wave height achieved by reef accretion, but also relates to the fact that SLR decreases island elevation above the still water level and therefore lowers the threshold energy required for overtopping. Even though vertical reef growth cannot prevent overtopping, the effectiveness in offsetting a rise in shoreline energy and significantly decreasing overwash volume provides substantial support for active coral reef restoration [START_REF] Clark | An evaluation of artificial reef structures as tools for marine habitat rehabilitation in the Maldives[END_REF][START_REF] Temmerman | Ecosystem-based coastal defence in the face of global change[END_REF][START_REF] Rinkevich | Rebuilding coral reefs: does active reef restoration lead to sustainable reefs?[END_REF] as a source of 'cost-effective' shoreline protection for island communities facing SLR [START_REF] Ferrario | The effectiveness of coral reefs for coastal hazard risk reduction and adaptation[END_REF].

Our results, elucidating changes in the full wave spectrum impacting shorelines and consequences for island overtopping, are critical for resolving the changing hazard-scape of islands and their ongoing habitability. However, reef islands are dynamic features that adjust their shoreline morphology in response to environmental variability across a range of temporal scales [START_REF] Mclean | Destruction or persistence of coral atoll islands in the face of 20th and 21st century sea-level rise?[END_REF] and future work is required to address the morphological adjustment of islands to SLR. However, the assumption of a static shoreline in this study provides a baseline 'worst case' assessment of the increasing exposure of island communities to wave hazards associated with increasing sea-level. Further, morphologic change on many developed islands is compromised by development pressure, environmental degradation, and histories of hard engineering [START_REF] Mclean | Destruction or persistence of coral atoll islands in the face of 20th and 21st century sea-level rise?[END_REF]. Communities on urbanized islands will be confronted with significant adaptation challenges over the next half This article is protected by copyright. All rights reserved.

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century if no action is taken to offset the current trajectory of climate change driven sea-level rise. [START_REF] Beetham | Field and numerical investigations of wave transformation and inundation on atoll islands[END_REF]. d) Reef and island morphology used in model simulations highlighting where wave data were analysed at the shoreline sensor and where overwash (OW) was quantified.

Figure captions

This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. Relative contribution of SS wave energy and IG energy to net spectral density at the shoreline with SLR. j-l) Rate of wave overwash (OW) across the island berm, with the overwash sealevel threshold (OWSL) shaded in dark grey.

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Figure 1 :

 1 Figure 1: Field setting. a) Location of Funafuti Atoll in the Pacific Ocean. b) Location of

Figure 2 :

 2 Figure 2: a) Comtemporary reef morphology. b-d) Hypothetical reef growth morphologies

Figure 3 :

 3 Figure 3: Boundary wave conditions. a-c) Pressure signal and d-f) spectral composition for

Figure 4 :

 4 Figure 4: Impact of sea-level rise with a static reef response on shoreline wave processes and

Figure 5 :

 5 Figure 5: Comparison between wave processes at the shoreline at high tide under static and

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Acknowledgements

This research was undertaken while Edward Beetham was being supported by a University of Auckland Doctoral Scholarship. All data associated with this research (including numerical model simulation files, input files and raw output files) are publically available from the Figshare repository doi: 10.17608/k6.auckland.5150140, url: https://figshare.com/s/eedcf0b1a8ab1b2b103a. The Green Naghdi solver used in this research is part of Basilisk and is freely available online at basilisk.fr.

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Tables Table 1: First year associated with overtopping at spring high tide under each simulated wave condition, assuming the reef and island undergo no morphological adjustment to SLR.

A range of SLR rates are presented, based on measurements form the instrumental record [START_REF] Becker | Sea level variations at tropical Pacific islands since 1950[END_REF] and various IPCC regional concentration pathways [START_REF] Church | Sea level change, in Climate Change 2013: The Physical Science Basis[END_REF].

Scenario


Baptiste Devyver

INDEX OF THE CRITICAL CATENOID

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The series of recent articles [START_REF] Fraser | The first Steklov eigenvalue, conformal geometry, and minimal surfaces[END_REF], [START_REF] Fraser | Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF] by A. Fraser and R. Schoen triggered a renewed interest for free boundary minimal surfaces. More specifically, Fraser and Schoen studied mainly free boundary minimal surfaces in the unit ball B n of R n , and in [START_REF] Fraser | The first Steklov eigenvalue, conformal geometry, and minimal surfaces[END_REF], they provide two interesting examples of those, namely the critical catenoid (a rescaled piece of the usual catenoid in R 3 ), and the critical Möbius band (a non-orientable free boundary minimal surface in B 4 ). They also constructed in [START_REF] Fraser | Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF] free boundary minimal surfaces in B 3 of genus 0 and with an arbitrary number of boundary components. Guided by an hypothetical analogy between (closed) minimal surfaces of S 3 and free boundary minimal surfaces of B 3 , P. Sargent [16] and independently L. Ambrozio, A. Carlotto and B. Sharp [START_REF] Ambrozio | Index estimates for free boundary minimal hypersurfaces[END_REF], have in particular given lower estimates for the Morse index of free boundary minimal surfaces in B 3 , in terms of their topology, similar to the one found by A. Savo [17] for minimal surfaces of the sphere S 3 . Actually, Savo's result also holds in higher dimensions, and the results [16], [START_REF] Ambrozio | Index estimates for free boundary minimal hypersurfaces[END_REF] are much more general than this since they deal with free boundary minimal surfaces in open sets of R n satisfying some convexity assumption (see [START_REF] Ambrozio | Index estimates for free boundary minimal hypersurfaces[END_REF]Theorem 10] for the strongest result so far obtained). Let us also mention the work [START_REF] Ambrozio | comparing the Morse index and the first Betti number of minimal hypersurfaces[END_REF], where Savo's results are in particular extended to closed minimal hypersurfaces inside rank one symmetric spaces. A heuristic that has been in the air for some time concerning the analogy between closed minimal surfaces of S 3 and free boundary minimal surfaces of B 3 is the following: since the Clifford torus is arguably the simplest non-trivial closed minimal surface in S 3 , and the critical catenoid is the simplest non-trivial free boundary minimal surface in B 3 , is it possible that the two be "analogous" in some sense? There are several celebrated characterizations of the Clifford torus: it is the unique minimal torus in S 3 (Lawson's conjecture, proved by S. Brendle [4]), it has minimal Willmore energy among all tori in S 3 (Willmore conjecture, proved by F. Coda Marques and A. Neves [8]), and finally, it has minimal index (equal to 5) among all non-totally geodesic minimal surfaces in S 3 (F. Urbano,[15]). Actually, Urbano's result has been important in Coda Marques and Neves' approach. It is very natural to ask whether a similar characterization to Urbano's for the Clifford torus holds as well for the critical catenoid. However, while it is quite easy to compute the index of the Clifford torus, computing the index of the critical catenoid is harder, due to the presence of the boundary and the more complicated equations. Our main goal in this article is to fill this gap and to show that minimal disks excluded, the critical catenoid has the smallest index possible among all orientable free boundary surfaces of the ball. More precisely, we show the following result: Theorem 1.1. Every orientable, free boundary minimal surface in the unit ball B 3 , is either a flat disk passing through the origin (and in this case has index 1), or has index at least 4. Moreover, the index of the critical catenoid is exactly 4, so that the latter lower bound for the index is attained.

Remark 1.2. The nullity of the critical catenoid can also be computed. According to [START_REF] Máximo | Free boundary minimal annuli in convex three-manifolds[END_REF]Prop. 6.15], it is equal to 2.

In the case of minimal surfaces of S 3 , according to F. Urbano [15], the Clifford torus is the unique minimal surface of index 5. We conjecture that this property also holds for the critical catenoid:

Conjecture 1.3. Let Σ be a free boundary minimal surface in the unit ball B 3 with index 4. Then, Σ is isometric to the critical catenoid.

Remark 1.4. While this article was in preparation, the author has been informed of related works by G. Smith and D. Zhou [18] on the one hand, and by H. Tran [START_REF] Tran | Index Characterization for Free Boundary Minimal Surfaces[END_REF] on the other, in which the index of the critical catenoid is also computed. Our proof and H. Tran's use the Steklov spectrum of the Jacobi operator, while G. Smith and D. Zhou's use the Robin spectrum. However all three proofs are different, and have been obtained independently.

The plan of this article is as follows: in Section 2, we set up the setting and recall well-known definitions. In Section 3, we present the critical catenoid and we introduce suitable coordinates. In Section 4, we study the Dirichlet problem for the stability operator on the catenoid, that is instrumental in our proof of Theorem 1.1. In Section 5, we prove the part of Theorem 1.1 pertaining to the critical catenoid (see Theorem 5.1). In Section 6, we finish the proof of Theorem 1.1, by proving the universal lower bound for the index (see Proposition 6.2). In Section 7, we present some partial results related to Conjecture 1.3.

During the time this research was carried out, the author was partially supported by the Natural Sciences and Engineering Research Council of Canada through a post-doctoral fellowship.

Free boundary minimal surfaces in the ball

Let us first recall the concept of a free boundary minimal hypersurface of the unit ball B n+1 = {x ∈ R n+1 ; ||x|| ≤ 1} of R n+1 . For the purpose of this article, a hypersurface (with boundary) of R n+1 is called admissible if it is smoothly embedded in B n+1 , and if its boundary lies in the unit sphere S n = ∂B n+1 . A free boundary minimal hypersurface in B n+1 is by definition a critical point of the n-volume functional, restricted to admissible hypersurfaces. Equivalently, it is an admissible hypersurface Σ n which is minimal, and which intersects orthogonally the unit sphere S n at its boundary. This last property is easily seen to be equivalent to the fact that at the boundary of Σ n , the exterior normal ν coincides with the position vector X. In this article, we shall be concerned with the index of these free boundary minimal hypersurfaces. We will assume that Σ n is oriented, and denote by N a smooth unit normal. The index of a free boundary minimal hypersurface in B n+1 can be defined as the (Morse) index of the quadratic form associated with the second variation of the n-volume functional, defined on the set of admissible hypersurfaces. It is well-known that if Σ is a free boundary minimal hypersurface, and Σ t , t ∈ [0, 1) is a one-parameter family of admissible, normal deformations of Σ (thus, Σ 0 = Σ), then

∂ ∂t t=0 Vol(Σ t ) = 0, ∂ 2 ∂t 2 t=0 Vol(Σ t ) = Σ (|∇u| 2 -|A| 2 u 2 ) - ∂Σ u 2 .
Here, u is a smooth, real function on Σ, such that the deformation Σ t occurs at t = 0 in the direction of the normal vector field uN . Also, A denotes the second fundamental form, and |A| 2 is the square of its norm, i.e. the sum of the squares of the principal curvatures. This leads one to consider the quadratic form

Q(u) = Σ (|∇u| 2 -|A| 2 u 2 ) - ∂Σ u 2 .
The quadratic form Q is naturally associated to a second-order differential operator, the Jacobi operator J = ∆ -|A| 2 (in this article, we shall take the convention that ∆ has non-negative spectrum). The (Morse) index of Σ, as a free boundary minimal hypersurface, is then defined as the maximal dimension of a vector space of smooth functions on Σ, on which Q is negative definite. The index has been extensively studied for closed hypersurfaces, in which case it is simply equal to the number of negative eigenvalues of the Jacobi operator. However, the presence of a boundary term in Q make things more difficult. It is well-known (see [START_REF] Chen | Minimal immersions of compact bordered Riemann surfaces with free boundary[END_REF]) that the index of Σ is equal to the number of negative eigenvalues of the Robin eigenvalue problem:

   Ju = 0 on Σ ∂u ∂ν = λu, However, in the free boundary case several other legitimate eigenvalue problems can be considered for J, in connection to the index (for example, Dirichlet, Steklov, Robin, etc...); and indeed, in order to compute the index of the critical catenoid, we will use the Steklov boundary conditions, and not the Robin ones.

To conclude this preliminary section, let us recall an interesting result, proved by Fraser and Schoen in [START_REF] Fraser | Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF]. For v a constant vector in R n+1 , denote v ⊥ := (N, v), which is a smooth function on Σ. Note also that Jv ⊥ = 0 for all v ∈ R 3 . That is, v ⊥ is a Jacobi field. Then, by [START_REF] Fraser | Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF]Theorem 3.1],

Q(v ⊥ ) = -2 Σ |v ⊥ | 2 .
If n = 2, the vector space {v ⊥ ; v ∈ R n+1 } has dimension 3, unless Σ is a flat disk. Thus, in particular, every free boundary minimal surface of B 3 , that is not a flat disk, has index at least 3. Our main result in this article implies that this lower bound can be improved to 4, and moreover 4 is optimal. Of course, a flat disk is a free boundary minimal surface of the unit ball if and only if it passes through the origin, and in this case it is not hard to check that it has index 1, by using polar coordinates and Fourier analysis.

The critical catenoid

In this section, we present the main protagonist of this article, namely the critical catenoid, and give some useful properties related to it. Let C a , a > 0, be the catenoid in R 3 parametrized by X a (t, θ) = (a cosh(t/a) cos(θ), a cosh(t/a) sin(θ), t) = a(cosh(s) cos(θ), cosh(s) sin(θ), s), s = t/a.

On C a , the unit normal is given by

N = - cos(θ) cosh(t/a) , - sin(θ) cosh(t/a)
, tanh(t/a) .

A simple computation shows that there exists a unique value of the parameter a, such that C a intersects the unit sphere orthogonally. For this value of a, the part of C a lying inside the unit ball is called the critical catenoid. In the above coordinates, it means that t ∈ [-aT, aT ], where T is the unique positive solution of the equation

T tanh(T ) = 1.
With this definition of T , the dilation parameter a has the value

a = 1 T cosh(T )
.

From now on and until Section 6, Σ will denote the critical catenoid as described above.

3.1.

A choice of coordinates. In the coordinates (s, θ), s = t/a, the metric on Σ writes

g = a 2 cosh 2 (s)(ds 2 + dθ 2 ),
with s ∈ [-T, T ] and θ ∈ [0, 2π). The Jacobi operator J is given by

Ju(s, θ) = - 1 a 2 cosh 2 (s) ∂ 2 u ∂s 2 + ∂ 2 u ∂θ 2 - 2 a 2 cosh 4 (s) u,
and the quadratic form Q writes

Q(u(s, θ)) = [-T,T ]×S 1 ∂u ∂s 2 + ∂u ∂θ 2 - 2 cosh 2 (s) u 2 dsdθ -1 T 2π 0 u 2 (T, θ) + u 2 (-T, θ) dθ
It will be convenient to conformally change the metric g on Σ, in order to see the resulting surface as a part of the unit sphere S 2 and hence use spherical coordinates. While not strictly necessary (our proof would work without it), it does simplify the Jacobi operator, and the equation for Jacobi fields become easier. Thus, let us consider the metric g, conformal to g, defined by g = (cosh(s)) -2 (ds 2 + dθ 2 ).

Since g = K g g, where K g is the Gauss curvature of g, it follows that g has Gauss curvature constant and equal to 1. Moreover, it is easily seen that (Σ, g) is isometric to an annulus inside the 2-dimensional sphere S 2 : indeed, let

ϕ = 2 arctan(e -s ),
then by use of standard trigonometric formulas, one computes that cosh -1 (s) = sin(ϕ).

Therefore, the metric g can be written as

g = dϕ 2 + sin 2 (ϕ)dθ 2 = g (S 2 ,can) ,
the canonical metric on S 2 written in spherical coordinates (ϕ, θ). The surface (Σ, g) is therefore isometric to the annulus Ω := {ϕ * ≤ ϕ ≤ π -ϕ * , 0 ≤ θ < 2π} of the unit sphere S 2 . Also, observe that tanh(s) = cos(ϕ).

Therefore, since -T ≤ s ≤ T , and T tanh(T ) = 1, one obtains

(θ, ϕ) ∈ Ω ⇔ - 1 T ≤ cos(ϕ) ≤ 1 T .
Concerning the quadratic form Q, using the conformal invariance of the Dirichlet energy in dimension 2, one can write Q as

Q(u) = Ω (|∇u| 2 -2u 2 ) - cosh(T ) T ∂Ω u 2 dσ,
where all the quantities (gradient, volume, etc) are computed in the canonical metric of S 2 . Thus, one sees that the operator naturally associated to Q in the metric g is

J = ∆ S 2 -2.
Hence, as promised, in the conformal metric g = g (S 2 ,can) the operator associated to Q has a particularly simple expression.

3.2. Jacobi fields. Recall that, by a slight abuse of notation, a smooth function u on Σ is called a Jacobi field if Ju = 0. For the critical catenoid, there are several interesting Jacobi fields that one can consider: one example that we have already mentioned in Section 2 is v ⊥ = (v, N ) for v a constant vector in R 3 . Interestingly, if one considers the canonical basis (v x , v y , v z ) of R 3 , one obtains three Jacobi fields v ⊥ x , v ⊥ y and v ⊥ z , that have the extra property of being Steklov eigenfunctions for J; that is, they solve a Steklov boundary value problem:

(3.1) Ju = 0, on Σ, ∂u ∂ν = λu, on ∂Σ, where ν is the exterior unit normal to Σ on ∂Σ (which is also equal to X, the position vector, by the free boundary condition). More precisely, by a straightforward computation, one checks:

Lemma 3.1. The function v ⊥
x , v ⊥ y and v ⊥ z are Steklov eigenfunctions for the Jacobi operator J, associated respectively to the eigenvalues -1, -1 and

1 sinh 2 (T ) .
Notice that by the Green formula, if u satisfies (3.1), then Q(u) < 0 if and only if λ < 1. This is the case for the functions v ⊥

x , v ⊥ y and v ⊥ z . It will also be useful for future reference to write down the boundary values of these functions; to this end, write ∂Σ = ∂Σ + ∪ ∂Σ -, with

∂Σ ± := {(s, θ) ; s = ±T, θ ∈ [0, 2π)}. Then, v ⊥ x = 1 cosh(T ) cos(θ), v ⊥ y = 1 cosh(T ) sin(θ), v ⊥ z = ±1 T .
According to [5, Propositions 2.1, 2.2], there are also three Jacobi fields coming from Killing (rotation) vector fields of R 3 , and one Jacobi field coming from varying the parameter a defining the catenoid. The former are v ⊥ (x,z) := (-z∂ x + x∂ z , N ), v ⊥ (y,z) := (-z∂ y + y∂ z , N ) and v ⊥ (x,y) := -y∂ x +x∂ y , N ), while the latter is ξ := -∂ ∂a X a (t, θ), N . By the fact that rotations with center at the origin preserve the unit ball and the definition of the quadratic form Q in terms of second variation of the area functional (see Section 2), it is obvious that

Q(v ⊥ (x,z) ) = Q(v ⊥ (y,z) ) = Q(v ⊥ (x,y) ) = 0. Explicitly, in coordinates, one has v ⊥ (x,z) = Λ(s) cos(θ), v ⊥ (y,z) = Λ(s) sin(θ), v ⊥ (x,y) = 0, with (3.2) Λ(s) = a s cosh(s) + tanh(s) cosh(s) .
It is easily checked that v ⊥ (x,z) and v ⊥ (y,z) are Steklov eigenfunctions for J, with associated eigenvalue 1. Concerning the Jacobi field ξ, one computes in coordinates that ξ = 1 -s tanh(s). From this computation, and recalling that s ranges from -T to T with T tanh(T ) = 1, one concludes that ξ is positive in the interior of Σ and vanishes on the boundary of Σ. The nonnegativity of ξ implies that it is a first eigenfunction of J with Dirichlet boundary conditions. If we denote by λ 1 (J) < λ 2 (J) ≤ • • • ≤ λ n (J) ≤ • • • the spectrum of J with Dirichlet boundary conditions, it follows that λ 1 (J) = 0 (thus, Σ is a maximal stable domain of a rescaled catenoid, for variations preserving the boundary). Of course, this is a well-known fact (see [5]).

3.3.

A Fourier decomposition. We introduce a natural Fourier decomposition for Jacobi fields: working in coordinates (ϕ, θ) on Σ, and assuming that u is a smooth, real function on Σ, one can write u(ϕ, •) as the sum of its Fourier series in θ:

(3.3) u(ϕ, θ) = a 0 (ϕ) + ∞ n=1 (a n (ϕ) cos(nθ) + b n (ϕ) sin(nθ)).
Using the well-known formula ∆ g = λ -2 ∆ g relating the Laplacian ∆ g of a metric g = λ 2 g conformal to g, one easily finds that

J = λ -2 J.
According to our computations in Section 3.1, J = ∆ (S 2 ,can) -2. Hence, Ju = 0 if and only if in (ϕ, θ) coordinates, (∆ S 2 -2)u(ϕ, θ) = 0. Recalling the expression of the Laplacian of the unit sphere in spherical coordinates,

-∆ = 1 sin(ϕ) ∂ ∂ϕ sin(ϕ) ∂ ∂ϕ + 1 sin 2 (ϕ) ∂ 2 ∂θ 2 ,
one finds that u is a Jacobi field if and only if for every n ≥ 0 (resp. n ≥ 1), the functions a n (ϕ) (resp. b n (ϕ)) are solutions of the following ODE, known in the literature as the associated Legendre equation with indices (1, n) (see for example [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]):

(3.4) - 1 sin(ϕ) ∂ ∂ϕ sin(ϕ) ∂ ∂ϕ + n 2 sin 2 (ϕ) -2 a n (ϕ) = 0, for ϕ ∈ [ϕ * , π -ϕ * ]
. This suggests to introduce the differential operators L n , defined by

(3.5) L n = - 1 sin(ϕ) ∂ ∂ϕ sin(ϕ) ∂ ∂ϕ + n 2 sin 2 (ϕ) -2 .
The operator L 0 actually corresponds to the "radial part" of the Jacobi operator J: more precisely, if u depends only on s (or, equivalently, on ϕ), then Ju = 0 if and only if L 0 u = 0. We observe that the equation L 0 a(ϕ) = 0 is a regular ODE on the interval [ϕ * , π -ϕ * ], and thus has a space of dimension 2 of solutions: explicitly, a basis of the space of solutions is provided by ξ and v ⊥ z . Consequently, if u is a Jacobi field, i.e. if Ju = 0, then its Fourier mode of order zero, a 0 (ϕ), is a linear combination of ξ and v ⊥ z . Concerning the equation L 1 a(ϕ) = 0, which is also a regular ODE, a basis of solution is obtained by looking at the expression of the Jacobi fields v ⊥

x and v ⊥ y on the one end, and v ⊥ (x,z)

and v ⊥ (y,z) on the other hand, in (ϕ, θ) coordinates. Indeed,

v ⊥ x = -χ(s) cos(θ), v ⊥ y = -χ(s) sin(θ), with χ(s) = cosh -1 (s) = sin(ϕ) satisfying L 1 χ = 0.
The other independent solution of L 1 a(ϕ) = 0 is obtained by writing the function Λ(s) from (3.2) as a function of ϕ rather than s; by a slight abuse of notation, we will also denote this function Λ(ϕ). Thus, if u is a Jacobi field, then its Fourier mode of order 1, a 1 (ϕ) cos(θ) + b 1 (ϕ) sin(θ), is a linear combination of the four Jacobi fields v ⊥

x , v ⊥ y , v ⊥ (x,z) and v ⊥ (y,z) . Altogether, for a Jacobi field u, the Fourier decomposition in the θ variable can be equivalently written as

(3.6) u(ϕ, θ) = av ⊥ z + bξ + αv ⊥ x + βv ⊥ y + γv ⊥ (x,z) + δv ⊥ (y,z) + ∞ n=2 (a n (ϕ) cos(nθ) + b n (ϕ) sin(nθ)),
where a, b, α, β, γ, δ are real numbers and a n (ϕ), b n (ϕ) are solutions of (3.4).

The Dirichlet problem

A crucial ingredient in our proof that the critical catenoid has index 4, is the solution of the Dirichlet problem for the stability operator J. However, since Σ is a maximal domain of stability, the operator J has a non-trivial kernel. Thus, bearing in mind the Fredholm alternative, we introduce a zero-flux condition as follows. For a smooth function u on Σ, we define its flux on the boundary:

Φ(u) = ∂Σ ∂u ∂ν .
We will say that u has zero flux at the boundary if Φ(u) = 0.

Lemma 4.1. Let u be a smooth function on ∂Σ. Then, the Dirichlet problem

   J û = 0 on Σ û = u on ∂Σ,
is solvable if and only if ∂Σ u = 0, and in this case it has a unique solution û with zero flux at the boundary.

Proof. Let ũ be a smooth extension of u to Σ (supported in a tubular neighborhood of ∂Σ).

Then, writing û = v + ũ and f = -J ũ, we see that solving the Dirichlet problem is equivalent to solving (4.1)

Jv = f on Σ, v = 0 on ∂Σ.
By the Fredholm alternative (see [9, Thm 6.2.4]), there exists a solution of (4.1) if and only if

Σ f w = 0,
for every w such that J * w = 0, w| ∂Σ = 0. Since J is self-adjoint, J * = J. Also, 0 being the bottom of the spectrum of J with Dirichlet boundary conditions, it has multiplicity one as a Dirichlet eigenvalue of J. Thus, (4.1) has a solution if and only if

Σ (J ũ)ξ = 0.
But by the Green formula,

Σ (J ũ)ξ -(Jξ)ũ = ∂Σ u ∂ξ ∂ν -ξ ∂ ũ ∂ν .
Taking into account that Jξ = 0, ξ| ∂Σ = 0 and ũ| ∂Σ = u, we obtain that (4.1) has a solution if and only if

0 = ∂Σ u ∂ξ ∂ν .
Using that ∂ξ ∂ν is constant on ∂Σ, we deduce that ∂Σ u = 0 is a necessary and sufficient condition for solving the Dirichlet problem. Concerning uniqueness, just note that if u 1 and u 2 are both solutions of the same Dirichlet problem, then u := u 1 -u 2 satisfies Ju = 0 and u| ∂Σ = 0, which implies that u is a multiple of ξ (and conversely). Since ∂Σ ∂ξ ∂ν = 0, it follows that if the Dirichlet problem is solvable then there is precisely one solution with zero flux.

Index of the critical catenoid

Our aim in this section is to compute the index of the critical catenoid:

Theorem 5.1. The Morse index of the critical catenoid in the unit ball B 3 is exactly 4.

The idea of the proof is to show that the vector space W generated by the constant function 1 and the Jacobi fields v ⊥ , v ∈ R 3 , is a 4-dimensional space on which Q is negative definite, and with the following property: if u is a smooth function on Σ such that for every w in W, Q(u, w) = 0 (in other words, if u is Q-orthogonal to W), then Q(u) ≥ 0. Then, that the index is 4 follows by a standard projection argument. In order to make this idea work, an important point will be to decompose u, thanks to the solution of the Dirichlet problem in Section 4, as the sum of a Jacobi field, and of a smooth function vanishing at the boundary of Σ. 5.1. Index is at least 4. Later, in Section 5, we will prove that every free boundary minimal surface in the unit ball of R 3 , that is not a flat disk, has index at least 4. However, in the case of the critical catenoid, there is an easy elementary argument, which we present now. Consider the Jacobi fields v ⊥

x , v ⊥ y and v ⊥ z . Recalling their boundary values (see Section 3.2), it is easily seen that v ⊥

x , v ⊥ y and v ⊥ z are pairwise L 2 -orthogonal when restricted to ∂Σ. We have already mentioned in Section 3.2 that v ⊥

x , v ⊥ y and v ⊥ z are Steklov eigenvalues for the operator J, that is they satisfy the boundary value problem (3.1). But by the Green formula, if Ju = 0 and

∂u ∂ν = λu, then Q(u, v) = ∂Σ v( ∂u ∂ν -u) = (λ -1) ∂Σ uv.
and so we conclude from the pairwise

L 2 -orthogonality of v ⊥ x , v ⊥ y and v ⊥ z on ∂Σ that 0 = Q(v ⊥ x , v ⊥ y ) = Q(v ⊥ x , v ⊥ z ) = Q(v ⊥ y , v ⊥ z )
. Also, if one considers the constant function 1 on Σ, one has obviously

Q(1) = - Σ |A| 2 -L(∂Σ) < 0,
and moreover by the above Green formula, and by the fact that

0 = ∂Σ v ⊥ x = ∂Σ v ⊥ y = ∂Σ v ⊥ z ,
one sees that

Q(1, v ⊥ x ) = Q(1, v ⊥ y ) = Q(1, v ⊥ z ) = 0.
From this discussion, we conclude that the quadratic form Q is negative definite on the vector space W generated by 1 and the v ⊥ , v ∈ R 3 . Since this is a 4-dimensional space, we conclude that the index of Σ is at least 4. 5.2. Index and Jacobi fields. Before proving the upper bound for the index, we need a crucial preliminary result, which states as follows:

Proposition 5.2. Let u be a Jacobi field on Σ, such that

0 = ∂Σ u = ∂Σ uv ⊥ , v ∈ R 3 .
Then,

Q(u) ≥ 0.
Proof. We work in (ϕ, θ) coordinates and use the Fourier series (3.6) in the θ variable, introduced in Section 3.3:

u(ϕ, θ) = av ⊥ z + bξ + αv ⊥ x + βv ⊥ y + γv ⊥ (x,z) + δv ⊥ (y,z) + ∞ n=2 (a n (ϕ) cos(nθ) + b n (ϕ) sin(nθ)),
where a, b, α, β, γ, δ are real numbers. We first claim that under the hypotheses of Proposition 5.2, a = α = β = 0. Indeed, one clearly has

0 = ∂Σ uv ⊥ z = a ∂Σ |v ⊥ z | 2 ,
hence a = 0. Next we show that α = 0. Clearly,

∂Σ uv ⊥ x = α ∂Σ |v ⊥ x | 2 + γ ∂Σ v ⊥ x v ⊥ (x,z) ,
all the other terms obviously vanishing by the well-known orthogonality properties of trigonometric functions. But recall that v ⊥ x = χ(ϕ) cos(θ) with χ even, whereas v ⊥ (x,z) = Λ(ϕ) cos(θ) with Λ odd. Therefore,

∂Σ v ⊥ x v ⊥ (x,z) = 0. Consequently, 0 = ∂Σ uv ⊥ x = α ∂Σ |v ⊥ x | 2 ,
and it follows that α = 0. A similar argument yields β = 0.

Next, it follows easily from the Green formula that the above Fourier decomposition of u is Q-orthogonal (to treat the two terms Q(v ⊥

x , v ⊥ (x,z) ) and Q(v ⊥ y , v ⊥ (y,z) ), use as above the fact that χ is even and Λ is odd). Hence, using that 0

= Q(ξ) = Q(v (x,z ) ⊥ ) = Q(v ⊥ (y,z) ), one gets Q(u) = ∞ n=2 Q(a n (ϕ) cos(nθ)) + Q(b n (ϕ) sin(nθ)).
For n ≥ 2, define a quadratic form Q n on smooth function f (ϕ), ϕ ∈ [ϕ * , π -ϕ * ], by requiring that

Q n (f (ϕ)) = Q(f (ϕ) cos(nθ)) = Q(f (ϕ) sin(nθ)).
More explicitly, Q n is given by the following formula:

2Q n (f (ϕ)) = π-ϕ * ϕ * |f ′ (ϕ)| 2 + n 2 sin 2 (ϕ) -2 f 2 (ϕ) sin(ϕ) dϕ - 1 T (f (π -ϕ * ) 2 + f (ϕ * ) 2 ).
Thus, (

Q(u) = ∞ n=2 Q n (a n (ϕ)) + Q n (b n (ϕ)). It is obvious that if n ≥ 2, Q n (f (ϕ)) ≥ Q 2 (f (ϕ)). 5.1)
The result of Proposition 5.2 is a direct consequence of the following claim:

Claim:

The quadratic form Q 2 is positive definite.

In order to prove this claim, we need a criterion for positivity of quadratic forms having a boundary term: Lemma 5.3. Let S be a quadratic form acting on functions f (ϕ), ϕ ∈ [a, b], defined as

S(f ) = b a (f ′ ) 2 + V f 2 m(ϕ)dϕ -α(f 2 (a) + f 2 (b)),
where m is a positive, smooth function, α is a real number, and

V is a real potential on [a, b]. Assume that there is a positive function h(ϕ), ϕ ∈ [a, b], such that 1 m d dϕ m d dϕ -V h ≤ 0,
and such that

-m(a) d dϕ ϕ=a (log h) > α, m(b) d dϕ ϕ=b (log h) > α.
Then, S is positive definite.

Proof of Lemma 5.3: since h is positive, one can write f (ϕ) in the form f (ϕ) = g(ϕ)h(ϕ). One then expands the term (f ′ ) 2 , to find that

S(f ) = b a g 2 (h ′ ) 2 + 1 2 (g 2 ) ′ (h 2 ) ′ + h 2 (g ′ ) 2 + V g 2 h 2 m -α(f 2 (a) + f 2 (b)).
Integrating by parts the term (g 2 ) ′ (h 2 ) ′ and using the assumptions on h, one finds

S(f ) = b a -1 m d dϕ m dh dϕ + V h hg 2 m + b a h 2 (g ′ ) 2 m + mg 2 h dh dϕ b a -α(f 2 (a) + f 2 (b)) ≥ b a h 2 (g ′ ) 2 m + f 2 (a) -m(a) d dϕ ϕ=a (log h) -α + f 2 (b) m(b) d dϕ ϕ=b (log h) -α , > 0,
and the result of Lemma 5.3 follows.

In order to prove the claim, one uses Lemma 5.3; for that, one needs to find a suitable positive solution h(ϕ) of (3.4) for n = 2. Letting x = cos(ϕ), the equation (3.4) for n = 2 becomes

(5.2) - d dx (1 -x 2 ) d dx + 4 1 -x 2 -2 a(x) = 0, - 1 T ≤ x ≤ 1 T .
It is easily checked that the function (1 -x 2 ) -1 is a positive solution to (5.2). Coming back to the variable ϕ, one obtains the positive solution

h(ϕ) = 1 sin 2 (ϕ) (= cosh 2 (s))
to (3.4) for n = 2. One now checks that the h satisfies the hypotheses of Lemma 5.3. We compute that

d log h dϕ = -2 cotan(ϕ).
For ϕ = ϕ * , this is equal to -2 tanh(T ) cosh(T ) = -2 sinh(T ). Since T = cosh(T ) sinh(T ) , we get that 2 sinh(T ) > cosh(T ) T , so

- 1 cosh(T ) d dϕ ϕ=ϕ * (log h) > 1 T .
One proves similarly that 1 cosh(T )

d dϕ ϕ=π-ϕ * (log h) > 1 T .
Thus, h satisfies the hypotheses of Lemma 5.3, and the claim follows.

5.3. Index is at most 4. With the result of Proposition 5.2 at hand, one can now finish the proof of Theorem 5.1 concerning the index of the critical catenoid. As mentioned before, the idea is to prove that Q(u) ≥ 0, as soon as 0

= Q(u, 1) = Q(u, v ⊥ x ) = Q(u, v ⊥ y ) = Q(u, v ⊥ z )
. Let us define an auxiliary function f = u + a1, where a is chosen so that

∂Σ f = 0. Since Q(v ⊥ , 1) = 0, v ∈ R 3 , one has Q(f, v ⊥ ) = 0, v ∈ R 3 . Furthermore, since Q(u, 1) = 0, Q(u) = Q(f ) -a 2 Q(1).
Since Q(1) < 0, we see that in order to conclude that ≥ 0, it is enough to show that Q(f ) ≥ 0. Let h be the unique Jacobi field with zero flux at the boundary, such that h| ∂Σ = f | ∂Σ (the existence of h is guaranteed by Lemma 4.1). Then, f = h + g, with g| ∂Σ ≡ 0. By the Green formula,

Q(h, g) = Σ (Jh)g + ∂Σ g ∂h ∂ν -h = 0.
Consequently,

Q(f ) = Q(h) + Q(g). Since g| ∂Σ ≡ 0, Q(g) = Σ (|∇g| 2 -|A| 2 g 2 )
, which is non-negative since λ 1 (J) ≥ 0, i.e. Σ is stable for J with Dirichlet boundary conditions (see Section 3.2). Also, by integration by parts,

0 = Q(f, v ⊥ ) = ∂Σ f ∂v ⊥ ∂ν -v ⊥ ,
and since v ⊥ x , v ⊥ y and v ⊥ z are Steklov eigenfunctions with eigenvalue different from 1, the condition 0

= Q(f, v ⊥ ), v ∈ R 3 is equivalent to 0 = ∂Σ f v ⊥ , v ∈ R 3 . Thus, 0 = ∂Σ h = ∂Σ hv ⊥ , v ∈ R 3 .
According to Proposition 5.2, one concludes that Q(h) ≥ 0, so Q(f ) ≥ 0. The proof of Theorem 5.1 is complete.

A general lower bound for the index

In this section, we prove a general lower bound for the index of a free boundary minimal hypersurface Σ n of dimension n in the unit ball B n+1 of R n+1 , n ≥ 2. In order to do so, we will first need the following lemma, which will also be used in the next section: Lemma 6.1. Let Σ n be a free boundary minimal surface in B n+1 , which is not a flat B n . Let Q be the quadratic form coming from the second variation of Σ n , and let S be the auxiliary quadratic form defined by

S(u) = Σ n |∇u| 2 - ∂Σ n u 2 , u ∈ C ∞ (Σ). Then, for all u ∈ C ∞ (Σ n ) \ {0}, Q(u) < S(u).
Moreover, S ≤ 0 on the vector space V spanned by 1 and {x v , ; v ∈ R n+1 }, where by definition

x v := (x, v) for v ∈ R n+1 .
Proof. Recall that Q is given by the formula

Q(u) = Σ n |∇u| 2 -|A| 2 u 2 - ∂Σ n u 2 , u ∈ C ∞ (Σ). Therefore, it is obvious that for u ∈ C ∞ (Σ n ), Q(u) ≤ S(u),
the inequality being strict unless |A||u| vanishes identically. We argue by contradiction: let u ∈ C ∞ (Σ n ) \ {0}, and assume that |A||u| vanishes identically. By definition, |A| vanishes at a point p, if and only if the Gauss map N : Σ n → S n has a critical point at p. Denote by E ⊂ Σ n the set of critical points of the Gauss map. The fact that |A||u| vanishes identically implies that u vanishes identically on Σ n \ E. Since u is continuous and not identically zero, Σ n \ E cannot be dense in Σ n , or equivalently, E must have an interior point p in Σ n . This implies that the Gauss map is constant in a neighborhood of p, hence that a small piece of Σ n is included in a ball B n ; by an argument involving the unique continuation principle for the minimal graph equation, Σ n must be a ball B n , which is a contradiction. Thus, we have proved that for every u ∈ C ∞ (Σ n ) \ {0}, the function |A||u| cannot vanish identically. As we have seen, this implies that Q(u) < S(u), and the proof of the first part of the lemma is complete.

Concerning the second part, we argue as follows. By [START_REF] Fraser | The first Steklov eigenvalue, conformal geometry, and minimal surfaces[END_REF], since Σ n is a free boundary minimal hypersurface in B n+1 , the functions x v are Steklov eigenfunctions with eigenvalue 1. That is, ∆ Σ n x v = 0 in the interior of Σ, and

∂x v ∂ν = x v on ∂Σ.
The variational characterization of Steklov eigenvalues (see [11, p.4014]) implies that

∂Σ n x v = 0
(this can also be checked directly using the Green formula with x v and 1). Finally, one checks that S(1) < 0, S(x v ) = 0, and by integration by parts,

S(1, x v ) = ∂Σ n ∂x v ∂ν -x v = 0.
Thus, S ≤ 0 on V, and this concludes the proof of the second part of the lemma.

We are now ready to prove a general lower bound for the index. Proposition 6.2. Let Σ n be a free boundary minimal surface in B n+1 , which is not a flat B n . Then, the index of Σ is at least n + 2.

Remark 6.3. This improves upon [12, Theorem 3.1], which implies that (under slightly stronger assumptions than Proposition 6.2) the index of Σ n is at least n + 1.

Proof. Recall the auxiliary quadratic form S from Lemma 6.1, defined as

S(u) = Σ n |∇u| 2 - ∂Σ n u 2 .
By the second part of Lemma 6.1, S ≤ 0 on the vector space V spanned by 1 and {x v ; v ∈ R n+1 }, where x v := (x, v). The first part of Lemma 6.1 then implies that Q is negative definite on V.

Let us show that V is of dimension n + 2. Otherwise, there is a = 0 and v ∈ R n+1 such that a + x v = 0 on Σ n . Then, Σ n is included in the hyperplane H defined by

H = {x ∈ R n+1 ; x, v = -a}.
This means that Σ n is included in the flat n-ball B n+1 ∩ H, which necessarily passes through the origin by the free boundary condition, i.e. a = 0. This contradicts the assumption on Σ n . Consequently, we have proved that V has dimension n + 2, and the index of Σ n is at least n + 2.

Free boundary minimal surfaces of index 4

To conclude this article, we present two results pertaining to Conjecture 1.3. We note that these results have been obtained independently by H. Tran [START_REF] Tran | Index Characterization for Free Boundary Minimal Surfaces[END_REF]. We start with the following Lemma, which will be employed in the proof of these two results. Lemma 7.1. If Σ is a free boundary minimal surface in B 3 with index 4, then σ 1 (Σ), the first Steklov eigenvalue for ∆, is equal to 1.

Proof. Assume by contradiction that σ 1 (Σ) is strictly less than 1, then there is a first eigenfunction f satisfying ∆f = 0, ∂f ∂ν = σ 1 f and

∂Σ f = 0, ∂Σ f x v = 0, v ∈ R 3 .
Let V be the vector space spanned by 1 and {x v ; v ∈ R n }, and denote by W the vector space generated by V and f ; it is 5-dimensional, since f is L 2 -orthogonal to V in restriction to ∂Σ.

Recall the auxiliary quadratic form S from Lemma 6.1, defined by

S(u) = Σ |∇u| 2 - ∂Σ u 2 .
It follows from the Green formula that for every g ∈ V, S(f, g) = (σ 1 -1) ∂Σ f g = 0.

Moreover, S(f ) = (σ 1 -1) ∂Σ f 2 < 0 A flat disk passing through the origin having index 1 = 4, Σ is not a flat disk. Lemma 6.1 implies that Q is strictly negative on W, which contradicts the fact that Q has index 4. Therefore, σ 1 (Σ) = 1.

Corollary 7.2. Let Σ be an oriented, free boundary minimal surface in B 3 with index 4. Then, the Jacobi operator on Σ with Dirichlet boundary conditions has first eigenvalue equal to zero, and in particular Σ is stable for perturbations fixing its boundary. More precisely, the normal to Σ can be chosen so that the function ξ := (X, N ) is a positive Jacobi field in the interior of Σ. If moreover ∂Σ is embedded, then Σ has genus zero.

Proof. The stability of Σ follows from the proof of [START_REF] Fraser | Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF]Prop. 8.1]; for the sake of completeness, let us recall the argument. By the well-known Fischer-Colbrie-Schoen/Allegretto-Moss-Piepenbrink lemma (see [10], [START_REF] Brendle | Embedded minimal tori in S 3 and the Lawson conjecture[END_REF]), since (X, N ) is a Jacobi field, it is enough to show that it does not vanish on Σ to conclude that it is a first eigenfunction for J with Dirichlet boundary conditions. By Lemma (7.1), we know that for every v ∈ R 3 \ {0}, x v := (X, v) is a first Steklov eigenfucntion of ∆. By the nodal theorem for Steklov eigenfunctions, x v has at most 2 nodal domains. According to S.Y. Cheng [START_REF] Chen | Eigenfunctions and nodal sets[END_REF], at a point p belonging to the zero set of x v , N nodal half-lines meet at p if and only if x v vanishes at p to order N . Since x v has at most 2 nodal domains, it follows that it vanishes only at order one on its zero set. Since this is true for any v ∈ R 3 \ {0}, it follows that any plane passing through the origin intersects Σ transversally. Thus, the affine tangent plane to an interior point cannot pass through zero. Assume now that ξ vanishes at an interior point p. Thus, X belongs to the tangent plane at p. But then, the line containing p and having direction X passes through the origin, and is in the affine tangent plane at p, which is a contradiction. Therefore, ξ does not vanish in the interior, and we can choose the normal so that it is positive everywhere.

Assume now that ∂Σ is embedded. Define a surface Σ by gluing a disk on each connected component of Σ and smoothing out the corners of the obtained surface. Then, the map p → p |p| is a local homeomorphism from Σ into S 2 . Thus, it is a covering map, and since S 2 is simply connected, it must be a global homeomorphism. So, Σ has genus zero.

An immediate consequence of the Lemma (7.1), together with [12, Theorem 6.6], is the following result: Corollary 7.3. Let Σ be an oriented, free boundary minimal surface in B 3 . Assume that Σ has index 4, and is topologically an annulus. Then, Σ is congruent to the critical catenoid.

Acknowledgments

The author is thankful to A. Fraser for introducing him to this problem, as well as for many interesting discussions. What is more, all the results contained in Sections 6 and 7 of this paper have been obtained in collaboration with A. Fraser.