Néstor Novoa

Frederic Justaud

Paul Hamon

Thierry Roisnel

Olivier Cador

Boris Le Guennic

Carolina Manzur

David Carrillo
email: david.carrillo@ucv.cl

Jean-René Hamon
email: jean-rene.hamon@univ-rennes1.fr

Frédéric Justaud

Doubly phenoxide-bridged binuclear copper(II) complexes with ono tridentate schiff base ligand: Synthesis, structural, magnetic and theoretical studies

Keywords: Copper(II), ONO Tridentate Schiff base ligand, magnetic properties, structure determination, EPR study, Ab initio calculation

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Introduction

In recent years, there has been continuing interest in using bridging ligands in the synthesis of polynuclear complexes of paramagnetic transition metal ions. In particular, ligands which contain potentially bridging phenoxo, alkoxo or hydroxo oxygen and nitrogen donor atoms have been widely employed to build up multinuclear copper(II) complexes [1][START_REF] Ruiz | Electronic Structure and Magnetic Behavior in Polynuclear Transition-Metal Compounds[END_REF][START_REF] Anbu | [END_REF][4][5].

Such symmetrically or asymmetrically dibridged complexes with µ-phenoxo, µ-alkoxo or µhydroxo di-copper(II) core have been the subject of a considerable amount of work in terms of correlating structure and magnetic properties [1,[START_REF] Ruiz | Electronic Structure and Magnetic Behavior in Polynuclear Transition-Metal Compounds[END_REF][START_REF] Crawford | [END_REF][8][9][10][START_REF] Kahn | Molecular Magnetism[END_REF]. Moreover, doubly phenoxo bridged binuclear Cu(II) complexes have also considerable interest as they provide the simplest case of magnetic interaction including only two unpaired electrons [START_REF] Ruiz | Electronic Structure and Magnetic Behavior in Polynuclear Transition-Metal Compounds[END_REF][START_REF] Gatteschi | Magnetostructural Correlations in Exchange Coupled Systems[END_REF][START_REF] Kahn | Molecular Magnetism[END_REF][START_REF] Launay | Electrons in Molecules: From Basic Principles to Molecular Electronics[END_REF]. And it is now established that the major factor controlling the singlet-triplet energy gap, J, between the metal centers is the Cu-O-Cu angle [8,[START_REF] Thompson | [END_REF][14][15]. Di-copper(II) derivatives are also of importance as precursors in the chemistry of supramolecular [16] and discrete molecular highnuclearity copper(II) complexes [17][18][19][20][21].

Diprotic Schiff bases are easily prepared by 1:1 condensation of appropriate salicylaldehyde or β-diketone reagents with appropriate amino alcohol substrates under mild conditions. Upon double deprotonation, they form a group of di-negative tridentate O,N,Odonor ligands that react readily with cupric salts to produce variable-nuclearity compounds depending on the coligand [19,22]. Thus, a N-donor monodentate coligand results in the formation of neutral mononuclear square planar ternary copper(II) complexes while square pyramidal Cu(II) compounds are obtained in the presence of a N-N chelate coligand [19,[23][24][25][26][27]. Moreover, the very popular bridging ligand 4,4'-bipyridine has been used also to construct homobinuclear copper(II) complexes [28], acting as a linear spacer between the mononuclear units, thus allowing the investigation of the intramolecular magnetic interaction between the magnetic centers [29].

We were interested in preparing neutral ternary mononuclear metal complexes of the type [LCu(II)(py)] with differently substituted di-anionic tridentate ONO Schiff base ligands (L 2-) and pyridine as ancillary ligand, as building blocks for the construction of novel selfassembled push-pull systems. However, it turned out that the synthesis using three closely related nitrogenous bases as coligand, leads to the formation of two different dimeric Schiff base complexes than those expected. Herein, we report on the synthesis, spectral and structural characterization, magnetic properties and ab initio study of both compounds formulated as [Cu 2 L 2 ] (1) and [(py-tBu) 2 Cu 2 L 2 ] (2), where H 2 L is the mono-condensation product of 1-anisyl-1,3-butanedione and 2-aminophenol (see Scheme 1). An EPR study of both complexes was also carried out.

Experimental

Materials

Manipulations of air-sensitive compounds were performed under Argon atmosphere using standard Schlenk techniques. Solvents were dried and distilled according to standard procedures [START_REF] Armarego | Purification of Laboratory Chemicals[END_REF]. 4-methoxyacetophenone, potassium tert-butoxide, 2-aminophenol, pyridine, 4-tert-butylpyridine, 4-picoline, Ethyl acetate (99%) and copper(II) nitrate trihydrate were purchased from commercial sources and used without further purification. Compound 1anisyl-1,3-butanedione was synthesized according to the literature procedure [START_REF] Popic | [END_REF].

Characterization and instrumentation.

Infrared spectra were recorded on a Perkin-Elmer, Model Spectrum One, FT-IR spectrophotometer as KBr disks in the 4000 to 400 cm -1 range. Elemental analyses were conducted on a Thermo-FINNIGAN Flash EA 1112 CHNS/O analyzer by the Microanalytical Service of the CRMPO at the University of Rennes 1, France. Electron paramagnetic resonance (EPR) spectra were recorded at 298 K with a Bruker EMX-8/2.7 (X-band) spectrometer. The temperature dependences of the magnetizations for powdered samples have been measured with a SQUID magnetometer (Quantum design MPMS-XL5) operating between 2 and 300 K at a constant field of 2 kOe. The experimental data have been corrected from the sample holder diamagnetism and the intrinsic diamagnetism estimated from the Pascal's tables [32]. Melting points were determined in evacuated capillaries on a Kofler Bristoline melting point apparatus and were not corrected.

Preparations

2.3.1. Preparation of [{4-MeO-C 6 H 4 -C(O)CH=C(CH 3 )N-o-C 6 H 4 -O}Cu] 2 (1)
A Schlenk tube was charged with a magnetic stir bar, 200 mg (1.04 mmol) of 1-anisyl-1,3butanedione, 114 mg (1.04 mmol) of 2-aminophenol and 10 mL of toluene, and the mixture was refluxed for 2 hours. After cooling, a yellow solid deposited. The precipitate was filtered off and washed with a petroleum ether: diethyl ether (1:1 v:v) mixture. Then, 292 mg (2.6 mmol) of potassium tert-butoxide and 2 mL of THF were added and the reaction mixture was stirred at r.t. for 20 min, upon which time it turned dark-red. Pyridine (160 μL, 2.08 mmol) was then added, before dropwise addition of a solution of Cu(NO 3 ) 2 • 3H 2 O (252 mg, 1.04 mmol) in 5 mL of THF. The stirring was continued overnight. The reaction was quenched with 15 mL of EtOH, releasing a dark-green microcrystalline precipitate that was filtered off and washed with cold EtOH (3 x 10 mL) and diethyl ether (4 x 10 mL). The dark-green solid residue was dried under vacuum for 2 hours and dissolved in dichloromethane. The solution was subjected to slow evaporation for 5 days, affording 300 mg (0.44 mmol, 85% Yield) of black microcrystals. A crystal from this crop was used for X-ray structure determination. M.p.

X-ray crystal structure determination

Well-shaped single crystals of 1 and 2 of suitable dimensions were coated in Paratone-N oil, mounted on a Kaptan loop and transferred to the cold gas stream of the cooling device.

Intensity data were collected at T = 150(2) K on a APEXII, Bruker-AXS diffractometer, Mo-Kα radiation (λ = 0.71073 Å), equipped with a bidimensional CCD detector and were corrected for absorption effects using multiscanned reflections. The two structures were solved by direct methods using the SIR97 program [33], and then refined with full-matrix least-square methods based on F 2 (SHELXL-97) [34] with the aid of WINGX program [35].

All non-hydrogen atoms were refined with anisotropic atomic displacement parameters. All the hydrogen atoms were placed in their geometrically idealized positions and constrained to ride on their parent atoms. A summary of the details about crystal data, collection parameters and refinement are documented in Table 1, and additional crystallographic details are in the

Computational details

DFT calculations [37] were carried out with the Amsterdam Density Functional (ADF)

program [38]. The Becke approximation for exchange [39] and the Perdew expression for correlation [40] (BP) has been chosen with the TZ2P basis set [41]. The optimized geometries of both compounds 1 and 2 were characterized as true minima on the potential energy surfaces using vibrational frequency calculations.

To compute exchange couplings in 1 and 2, Complete active space self-consistent field (CASSCF) calculations [42], including two electrons in two molecular orbitals (MOs), have been performed using the MOLCAS 7.6 package [43] to generate a reference space (CAS [START_REF] Ruiz | Electronic Structure and Magnetic Behavior in Polynuclear Transition-Metal Compounds[END_REF][START_REF] Ruiz | Electronic Structure and Magnetic Behavior in Polynuclear Transition-Metal Compounds[END_REF]). The dynamical correlation effects were then incorporated by using the dedicated difference configuration interaction (DDCI) method [44] implemented in the CASDI code [45]. With this approach, one concentrates on the differential effects rather than on the evaluation of the absolute energies. DDCI1 (i.e. CAS+S) involves one hole and one particle (1h, 1p, 1h1p) single excitations on the full active space. DDCI2 also accounts for the two holes or two particles diexcitations (2h, 2p). Finally, the two holes/one particle (2h1p) and one hole/two particles (1h2p) excitations are taken into account in DDCI3 (i.e. CAS+DDCI).

Since

Results and Discussion

Synthesis and characterization

The

Infrared spectral study

The solid-state FT-IR spectra of 1 and 2 were assigned on the basis of frequency calculations on DFT optimized geometries of 1 and 2 [37][38][39][40][41]. Table 2 gathers both computed and experimental major frequencies and their respective attributions, showing a good agreement between the calculated and observed spectral data. The most salient feature found in the spectra of 1 and 2 is the characteristic strong intensity bands in the ranges 1587-1466 and 1604-1584 cm -1 , respectively, assigned to the

(C ••• C), (C ••• N), (C ••• O)

Description of the molecular and crystal structures

Perspective views of binuclear compounds 1 and 2, including the atom labeling scheme, are shown in Figs. 1 and2, while selected bond lengths and angles are listed in Table 3. Compound 1 crystallizes in the monoclinic system with space group P2 1 /c, while complex 2 crystallizes in the triclinic crystal system Pī space group. In both cases, the asymmetric unit consists of a neutral centrosymmetric bis(µ-phenoxo)di-copper(II) complex with the two halves of the dimeric unit related by a crystallographic inversion center in the middle of the four membered Cu 2 O 2 core.

In [19,20,24,48,49]. In compound 1, the dinuclear unit presents two tetracoordinated cupric ions in a slightly distorted square-planar environment (Fig. 1), with the coordination sphere formed by the deprotonated amide nitrogen atom and the carbonyl and phenoxo oxygen atoms of the acyclic Schiff base ligand, and the bridging oxygen atom of the second half-unit. The four bond lengths span the range 1.8749( 14)-1.9579( 13) Å (Table 3). The sum of the angles around the Cu(II) center is 359.97°, indicating a slightly distorted square-planar geometry around the metal atom. The diagonal angles are, indeed, found to be 176. By contrast, the binuclear unit of 2 exhibits two pentacoordinated Cu(II) centers in a square pyramidal (4+1) coordination geometry (Fig. 2). The basal plane consists of the same tridentate ONO-donor set of atoms as in 1 and the nitrogen atom of the monodentate 4-tertbutylpyridine, while the apical site is occupied by the phenoxido oxygen atom of another ONO-tridentate dianion that is basal to the second copper(II) metal ion of the dimer. The copper atom is deviated by 0.1022 Å from the basal plane due to the formation of the axial bond. The bond distances of the donor atoms of the tridentate Schiff base ligand to the central metal atom are in the range 1.9155 [START_REF] Kahn | Molecular Magnetism[END_REF] to 1.9729(13) Å, and are typical for Cu(II) complexes [19,20,[23][24][25][26][27]50,51]. The Cu(II)-N py bond distance of 2.0467(13) Å is slightly longer than those observed in related ternary square-planar Cu(II) complexes (1.90-1.96 Å) [23][24][25][26][27]. Such a lengthening may explain an easy dissociation of less donating and less sterically demanding monodentate N-donor ligand leading to the formation of square planar dimeric complex 1, as observed here with pyridine and 4-picoline. Obviously, the Cu(1)-O(1') axial bond length (2.445(3) Å) is much longer than the equatorial distances which may be ascribed to Jahn-Teller distortion. The Addison parameter (τ) [52] which is an index of distortion from the square-pyramidal to the trigonal-bipyramidal geometry is calculated to be 0.072, indicating that the pentacoordinated geometry is very little distorted from a perfectly square pyramid (τ = 0). The dihedral angle between the bridging Cu 2 O 2 plane and the mean basal plane of the copper square pyramid is 80.713(4)°. Thermal ellipsoids are drawn at 60% probability.

In both complexes, the fused five-and six-membered heterometallacycles formed upon the Schiff base condensation of anisoylacetone with aminophenol and subsequent chelation of the Cu(II) ion, are essentially co-planar. Moreover, the six-membered metallacycle is also co-planar with the anisyl substituent with dihedral angles of 3.202(3) and 0.868(2)° in 1 and 2, respectively, and of 25.60(3)° with the 4-tert-butylpyridine ligand in 2.

In addition, interatomic distances and angles (Tables 3 and4) are indicative of substantial  delocalization of the electron density through the chelate rings.

EPR spectroscopy

The X-band EPR spectra of compounds 1 and 2 have been recorded in fluid solutions and as powdered samples at room temperature (298 K). The spectra of compound 1, recorded on a polycrystalline powder, do not show any EPR signal. In contrast, EPR spectrum of 2, obtained in the polycrystalline state at room temperature (Fig. 3), exhibits unresolved broadband resonances, due to exchange coupling between the copper(II) ions, at g = 2.071 with a H peak-to-peak values of 71.5 G. Moreover, the spectrum showed a weak M S = ±2 half field transition (Fig. 3, insert) at g = 4.181, thus giving unequivocal proof of the existence of the triplet state (S = 1) in the solid.

Magnetic properties

The magnetic properties of compounds 1 and 2 have been measured from room temperature to 2 K using a constant applied field of 2 kOe. On cooling a powdered sample of 1 χ M T (χ M the magnetic molar susceptibility and T the temperature in Kelvin)

decreases continuously from 0.375 cm 3 K mol -1 down to almost zero (0.02 cm 3 K mol -1 ). This

Theoretical investigations

To complete magnetic measurements, ab initio calculations of the magnetic interactions in 1 and 2 have been performed at the CAS[2,2]SCF/DDCI level (see computational details). Such a computational approach usually gives excellent agreement with respect to experimental findings [57], even for through-space interactions such as in 2 [58].

Of course, DFT (within the broken symmetry (BS) approximation) could have been used instead of CASSCF/DDCI approach [59]. However, major drawbacks, among them the spin contamination of the BS wavefunction and the choice of the exchange-correlation functional, explain the present choice of wavefunction-based multireference approaches. Results are gathered in Table 4 and the magnetic orbitals of 1 are given in Fig. 7. At the best level of calculation, i.e. CAS+DDCI, the strong antiferromagnetic behaviour of 1 is recovered whereas no interaction is calculated in the case of 2 due to a poor through-space overlap between the magnetic orbitals. Both results confirm the experimental data although in the case of 1, slight discrepancy remains between the measured and calculated exchange couplings.

Table 4 Calculated exchange coupling constant J (cm -1 ) in model complexes of 1 and 2

Conclusion

In this report, we have shown that treatment of the dianionic ONO-tridentate ligand derived from the Schiff base condensation of 1-anisyl-1,3-butanedione and 2-aminophenol, with Cu(II) ions leads to the dinuclear complex 1 when the reaction is carried out in the presence of pyridine or 4-picoline, while the same reaction carried out in the presence of 4tert-butylpiridine results in the formation of the binuclear species 2. Both compounds have been authenticated by single crystal X-ray diffraction analysis that showed that dimeric complex 1 presents two tetracoordinated Cu(II) centers in a square planar environment and that dimer 2 exhibits two pentacoordinated Cu(II) centers with a square pyramidal coordination sphere. The two halves of the dimeric unit are related through a planar Cu 2 O 2 rhombic core in both cases. EPR study in fluid solutions indicates that the dimeric structure of 1 and 2 is destroyed upon dissolution. In the solid-state, 1 is EPR silent whereas both MS = ±1 and MS = ±2 transitions are observed for 2. The variable temperature magnetic measurements reveal that dimer 1 exhibits a strong antiferromagnetic interaction due to the coplanar disposition of the Cu(II) metal ions and acute Cu-O-Cu angle (102°), whereas dimer 2 shows negligible antiferromagnetic coupling caused by the basal-apical phenoxido oxygen bridging mode. The magnetic findings have been corroborated by theoretical Ab initio calculations.

Appendix A. Supplementary data

Scheme 1

 1 Scheme 1 Preparation of complexes 1 and 2

  stretching vibrations of the Schiff base skeleton, suggesting that the imine nitrogen and carbonyl oxygen atoms coordinate to the Cu(II) metal ion. The weak bands observed at 3049 and 3040 cm -1 for 1 and 2, respectively, are attributed to the aromatic ν(C-H) vibrations, while the bands found at 2932 and 2830 cm -1 in the spectrum of 1 and at 2926 and 2864 cm -1 in that of 2, are characteristic of aliphatic ν(C-H) vibrations. The observed medium intensity band at 1244 and 1242 cm -1 for 1 and 2, respectively, are due to the ν asym vibration mode of the CH 3 -O-aryl group. The deformation modes of the C-H bonds in 1 and 2 show up respectively at 772 and 734 cm -1 . Thus, the infrared spectral data are consistent with the structural features of both compounds (see below Section 3.3).

Fig. 1

 1 Fig. 1 Molecular structure of complex 1 showing partial atom numbering scheme. Hydrogen atoms are omitted for clarity. Thermal ellipsoids are drawn at 60% probability.

  42(6)° [O(1)-Cu(1)-O(2)] and 161.35(6)° [N(1)-Cu(1)-O(1')]. Those values deviate somewhat from idealized transoid angles of 180° expected for a perfectly square-planar compound. Deviations of the coordinating O(1),O(1'), O(2) and N(1) atoms around Cu(II) center from the least-square mean planes through them are 0.0224[START_REF] Kahn | Molecular Magnetism[END_REF], -0.0198(9), 0.0166(8) and -0.0192(9) Å, respectively, and that of the Copper atom from the same plane is 0.0290(6) Å.

Fig. 2

 2 Fig. 2 Molecular structure of complex 2 showing partial atom numbering scheme and apical short contact Cu-O interactions in dashed-line. Hydrogen atoms are omitted for clarity.

Fig. 3

 3 Fig. 3 Experimental EPR spectrum of complex 2 recorded in the solid state at 298 K with an expansion of the half-field region.

Fig. 4 X

 4 Fig. 4 X-band EPR spectrum of 2 recorded in CH 2 Cl 2 : 1,2-C 2 H 4 Cl 2 (1:1 v:v) mixture at 298 K.

Fig. 5 1 (Fig. 6 )

 516 Fig. 5 Temperature dependence of χ M T for compound 1 (empty circles) with the best-fitted

Fig. 5 )Fig. 6

 56 Fig.5). Compound 2 shows a slightly distorted square-pyramidal geometry, thus the magnetic interactions that are mediated by apical-equatorial bridges are expected to be negligible as the dx 2 -y 2 and dz 2 orbitals of the Cu(II) centers are orthogonal to each other (see below Section 3.6), and the axial Cu-O bond is significantly longer (0.5203 Å) than the equatorial ones.

Fig. 7

 7 Fig. 7 Magnetic MOs extracted from a CAS[2,2]SCF calculation over the triplet state.

Table 1

 1 Crystallographic data, details of data collection and structure refinement parameters for compounds 1 and 2

			1	2
	Empirical formula	C 34 H 30 Cu 2 N 2 O 6	C 52 H 56 Cu 2 N 4 O 6
	Formula mass, g mol -1	689.68	960.09
	Collection T, K	150(2)	150(2)
	Crystal system	Monoclinic	Triclinic
	space group	P2 1 /c	P ī
	a (Å)		9.0297(2)	9.4434(3)
	b (Å)		14.3356(4)	11.3431(4)
	c (Å)		11.4572(3)	11.4661(4)
	 (°)		90	101.758(2)
	 (°)		109.8300(10)	109.5860(10)
	γ (°)		90	92.739(2)
	V (Å 3 )		1395.15(6)	1123.86(7)
	Z		2	1
	D calcd (g cm -3 )	1.642	1.419
	Crystal size (mm)	0.15 x 0.07 x 0.04	0.41 x 0.11 x 0.06
	Crystal color	black	black
	F(000)		708	502
	abs coeff (mm -1 )	1.577	1.002
	 range (°)	3.41 to 27.48	2.96 to 27.48
	range h,k,l	-11/11, -17/18, -12/14	-12/12, -13/14, -14/14
	No. total refl	12339	18161
	No. unique refl	3183	5123
	Comp. to  max (%)	99.6	99.4
	Max/min transmission	0.939/0.856	0.942/0.807
	Data/Restraints/Parameters	3183/0/201	5123/0/294
	Final R	R 1 = 0.0285	R 1 = 0.0286
	[I>2(I)]	wR 2 = 0.0691	wR 2 = 0.0712
	R indices (all data)	R 1 = 0.0378	R 1 = 0.0331
			wR 2 = 0.0736	wR 2 = 0.0737
	Gof / F	2	1.016	1.055

Largest diff. Peak/hole (eÅ -3 ) 0.430/-0.324 0. 365/-0. 406

Table 2

 2 Computed and experimental infrared frequencies a for compounds 1 and 2

		Computed	1 b	2 b
	ν(C-H aryl)	3089(w)	3049(w)	3040(w)
	ν(C-H, -OCH 3 )	2992(w)	2932(w)	2926(w)
	ν sym (CH 3 )	2949(w)	2830(w)	2864(w)
	ν(C … O), ν(C … N), ν(C … C) 1587(s)-1466(s)	1581(s)-1485(s)	1604(s)-1584(s)
	ν asym (C-H, C 6 H 4 -O-CH 3 )	1240(m)	1244(m)	1242(m)
	(C-H)	719(m)	772(m)	734(m)

a in cm -1 . b recorded as KBr disk.

Table 3

 3 Selected bond distances (Å) and angles (°) for compounds 1 and 2

		1	2
		Bond distances
	Cu(1)-O(1)	1.9579(13)	1.9247(11)
	Cu(1)-N(1)	1.9183(16)	1.9729(13)
	Cu(1)-O(2)	1.8749(14)	1.9155(11)
	Cu(1)-X a	1.9579(13)	2.0467(13)
		Angles
	O(1)-Cu(1)-O(2)	176.42(6)	174.93(5)
	O(1)-Cu(1)-N(1)	84.30(6)	84.60(5)
	O(2)-Cu(1)-N(1)	97.76(6)	95.20(5)
	N(1)-Cu(1)-X a	161.35(6)	162.66(5)
	O(1)-Cu(1)-X a	77.05(6)	90.86(5)
	O(2)-Cu(1)-X a	100.86(6)	90.78(5)

a X = O(1) #1 for 1, X = n(2) for 2; #1 Symmetry transformations used to generate equivalent atoms: -x+1,-y,-z+1.

Acknowledgments

The authors thank Drs. M. Fuentealba (PUC Valparaíso) and P. Jehan (CRMPO, Rennes) for helpful assistance and discussion with DFT calculations and HRMS measurements, respectively. This research has been performed as part of the Chilean-French International Associated Laboratory for "Inorganic Functional Materials'' (LIAMIF-CNRS N°836). Financial support from the Fondo Nacional de Desarrollo Científico y Tecnológico [FONDECYT (Chile), grant no. 1130105 (D.C., C.M. and M. F.)], the Vicerrectoría de Investigación y Estudios Avanzados, Pontificia Universidad Católica de Valparaíso, Chile (D.C., C.M. and M. F.), the CNRS and the Université de Rennes 1 is gratefully acknowledged. N.N. Thanks the CONICYT (Chile) and BECAS-CHILE for support of a graduate and Joint Supervision Scholarship (PUCV / UR1) and the Rennes Metropole for the 2013 scholarship awarded doctoral internship.


Nicolas Tholozan
email: tholozan@unice.fr

deforming closed anti-de Sitter 3-manifolds

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Introduction 0.1. Closed AdS 3-manifolds. An anti-de Sitter (AdS) manifold is a smooth manifold equipped with a Lorentz metric of constant negative sectional curvature. In dimension 3, those manifolds are locally modelled on PSL(2, R) with its Killing metric, whose isometry group identifies to a Z/2Z × Z/2Z extension of PSL(2, R) × PSL(2, R) acting by left and right translations, i.e.

(g 1 , g 2 ) • x = g 1 xg -1 2 for all (g 1 , g 2 ) ∈ PSL(2, R) × PSL(2, R) and all x ∈ PSL(2, R).

The specific interest for anti-de Sitter structures on 3-manifolds takes its roots in the geometrization program of Thurston. Following Thurston's classification of the eight Riemannian geometries in dimension 3, Scott [START_REF] Scott | The geometries of 3-manifolds[END_REF] proved that a large class of closed 3-manifolds could be modelled on the sixth geometry, that is, PSL(2, R) with a left invariant Riemannian metric that is also PSO(2)-invariant on the right. In particular, those manifolds carry an AdS structure, called standard. At first, it was conjectured that all closed AdS 3-manifolds had to be standard (the conjecture still holds in higher dimension, see [START_REF] Zeghib | On closed anti-de Sitter spacetimes[END_REF]). However, it turned out progressively that the space of AdS structures on closed 3-manifolds is much richer than the space of standard structures.

First, Goldman [START_REF] William | Nonstandard Lorentz space forms[END_REF] noticed that, in some cases, a standard structure could be deformed into a non-standard one. Namely, he proved that, if S is a closed surface of negative Euler characteristic, j : π 1 (S) → PSL(2, R) a discrete and faithful representation and ρ : π 1 (S) → Date: September 22, 2014. The author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network).

PSL(2, R) a representation with values in a 1-parameter subgroup that is sufficiently close to the trivial representation, then the embedding (j, ρ)(π 1 (S)) = {(j(γ), ρ(γ)), γ ∈ π 1 (S)} ⊂ PSL(2, R) × PSL(2, R) acts properly discontinuously and cocompactly on PSL(2, R). (This was generalized later by Kobayashi [START_REF] Kobayashi | Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds[END_REF].) The quotient is nonstandard whenever the image of ρ is not included in a compact subgroup.

Recall that a Lorentz metric on a manifold possesses a geodesic flow, and that the metric is called complete if its geodesics run for all time. In dimension 3, a complete AdS manifold is a quotient of PSL(2, R) by a subgroup of PSL(2, R) × PSL(2, R) acting properly discontinuously on PSL(2, R). In [START_REF] Kulkarni | 3-dimensional Lorentz space-forms and Seifert fiber spaces[END_REF], Kulkarni and Raymond addressed the question of describing all complete AdS 3-manifolds. Their first assertion is that complete compact AdS 3-manifolds have finite level, that is, they are actually quotients of a finite cover of PSL(2, R). (A correct proof was given by Salein in his thesis [START_REF] Salein | Variétés anti-de Sitter de dimension 3[END_REF].) Therefore, closed AdS 3-manifolds are cyclic coverings of a quotient of PSL(2, R) by the action of a discrete subgroup of PSL(2, R) × PSL(2, R). Kulkarni and Raymond then proved that all such quotients look like Goldman's examples in a certain sense.

Theorem (Kulkarni-Raymond, [START_REF] Kulkarni | 3-dimensional Lorentz space-forms and Seifert fiber spaces[END_REF]). Let Γ be a discrete group acting properly discontinuously and cocompactly on PSL(2, R) via a faithful representation h : Γ → PSL(2, R) × PSL(2, R). Assume moreover that Γ is torsion-free. Then Γ is the fundamental group of a closed surface of negative Euler characteristic and h = (j, ρ) , with either j or ρ discrete and faithful. Remark 0.1. This theorem was generalized by Kobayashi [START_REF] Kobayashi | On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups[END_REF] and Kassel [START_REF] Kassel | Proper actions on corank-one reductive homogeneous spaces[END_REF] to quotients of any rank 1 semisimple Lie group G under the action of some subgroup of G × G by left and right translations.

Remark 0.2. Contrary to the Riemannian setting, a Lorentz metric on a closed manifold may not be complete. However, Klingler proved later [START_REF] Klingler | Complétude des variétés lorentziennes à courbure constante[END_REF], generalizing Carrière's theorem [START_REF] Carrière | Autour de la conjecture de L. Markus sur les variétés affines[END_REF], that closed Lorentz manifolds of constant curvature are complete. Hence the completeness assumption in Kulkarni-Raymond's work can be removed.

Recall that, as an application of Selberg's lemma [START_REF] Selberg | On discontinuous groups in higher-dimensional symmetric spaces[END_REF], if there exists a faithful representation of Γ into PSL(2, R)×PSL(2, R), then Γ admits a torsion-free finite index subgroup. Therefore, Kulkarni-Raymond's theorem describes all closed AdS 3-manifolds up to a finite cover. A consequence of their theorem is that any closed AdS 3-manifold is finitely covered by a circle bundle over a surface, and is itself a Seifert fiber space.

We now fix a closed oriented surface S of negative Euler characteristic χ(S). A given pair (j, ρ) of representations of π 1 (S) into PSL(2, R) does not necessarily induce a proper action of π 1 (S) on PSL(2, R). For instance, if ρ(γ) is conjugate to j(γ) for some γ = 1, then the infinite subgroup generated by γ acts via (j, ρ) with a fixed point. A pair (j, ρ) is called admissible if (j, ρ)(π 1 (S)) acts properly discontinuously and cocompactly on PSL(2, R).

Let (j, ρ) be an admissible pair. One easily checks that (ρ, j) is also admissible. If σ denotes the outer automorphism of PSL(2, R) given by conjugation by 1 0 0 -1 , then (σ•j, σ•ρ) is also admissible. Finally, for any g 1 , g 2 ∈ PSL(2, R), the pair (C

g 1 • j, C g 2 • ρ)
is still admissible (where C g i denotes the conjugation by g i on PSL(2, R)). We will denote by Adm(S) the quotient of the space of admissible pairs of representations of π 1 (S) by these transformations. The main objective of this article is to describe the topology of Adm(S) and, in particular, classify its connected components. This will answer one of the many interesting questions of the recent survey: Some open questions in anti-de Sitter geometry ([2], section 2.3).

Recall that, by the work of Goldman [START_REF] William | Discontinuous groups and the Euler class[END_REF], connected components of the space of representations of π 1 (S) into PSL(2, R) are classified by an integer called the Euler class. It can take any value between χ(S) and -χ(S) and is extremal (i.e. equal to ±χ(S)) if and only if the representation is Fuchsian, that is, discrete and faithful. The postcomposition with the outer automorphism σ exchanges the components of extremal Euler class. Therefore, up to switching the factors and postcomposing with σ, we can assume that any admissible pair (j, ρ) has j Fuchsian of Euler class -χ(S). From now on, we will consider Adm(S) as the space of admissible pairs (j, ρ) with j of Euler class -χ(S) modulo the action of PSL(2, R)× PSL(2, R) by conjugation.

If j is Fuchsian and ρ takes values into a compact subgroup of PSL(2, R), then the pair (j, ρ) is clearly admissible, and the quotient is standard. According to [START_REF] William | Nonstandard Lorentz space forms[END_REF], there are some non-standard admissible pairs with ρ in the connected component of the trivial representation. Kulkarni and Raymond believed that all non-standard structures could be obtained by deforming standard ones. It was proven wrong by Salein. Let us denote by H 2 the Poincaré half-plane and by g P the Poincaré metric on H 2 , of curvature -1. The group PSL(2, R) acts on H 2 by homographies and identifies with the group of orientation preserving isometries Isom + (H 2 , g P ). Salein noticed [START_REF] Salein | Variétés anti-de Sitter de dimension 3 exotiques[END_REF] that a sufficient condition for a pair (j, ρ) to be admissible is that j stricty dominates ρ, in the sense that there exists a (j, ρ)-equivariant map from (H 2 , g P ) to (H 2 , g P ) which is λ-Lipschitz for some λ < 1. As a consequence, he constructed admissible pairs (j, ρ) with ρ of any non-extremal Euler class. In particular, those admissible pairs cannot be continuously deformed into standard ones. Lastly, Kassel proved in [START_REF] Kassel | Quotients compacts d'espaces homogènes réels ou p-adiques[END_REF] that Salein's sufficient condition for admissibility is also necessary:

Theorem (Kassel). Let S be a closed surface of negative Euler characteristic and j, ρ two representations of π 1 (S) into PSL(2, R), with j Fuchsian. Then the pair (j, ρ) is admissible if an only if there exists a (j, ρ)-equivariant map from (H 2 , g P ) to (H 2 , g P ) that is λ-Lipschitz for some λ < 1. 0.2. Dominated representations. Kassel's criterion for admissibility raises many questions that may turn out to be interesting beyond the scope of 3-dimensional AdS geometry. Indeed, one can extend the definition of domination to a more general setting. Consider a closed surface S of negative Euler characteristic, and a contractible Riemannian manifold (M, g M ). Let j : π 1 (S) → PSL(2, R) be a Fuchsian representation, and ρ a representation of π 1 (S) into Isom(M, g M ). Since M is contractible, there always exists a smooth (j, ρ)-equivariant map from H 2 to M . Since S is compact, this map is Lipschitz. We can thus define the minimal Lipschitz constant

Lip(j, ρ) = inf λ ∈ R + | ∃f : H 2 → M (j, ρ)-equivariant and λ-lispchitz .
We then say that j strictly dominates ρ if Lip(j, ρ) < 1. 0.2.1. Length spectrum. A related notion of domination comes from the comparison of the length spetra of j and ρ. Recall that if g is an isometry of a metric space (M, d), the translation length of g is the number l(g) := inf

x∈M d(x, g • x) .
The length spectrum of a representation ρ : π 1 (S) → Isom(M ) is the function that associates to γ ∈ π 1 (S) the translation length of ρ(γ). We say that the length spectrum of j strictly dominates the length spectrum of ρ if there exists a constant λ < 1 such that for all γ ∈ π 1 (S),

l(ρ(γ)) ≤ λ l(j(γ)) .
It is straightforward that if a Fuchsian representation j strictly dominates ρ, then the length spectrum of j strictly dominates the length spectrum of ρ. The converse is highly non-trivial, but happens to be true in the cases we are interested in, where the target space is a smooth complete simply connected Riemannian manifold of sectional curvature bounded above by -1 (what we will call for short a Riemannian CAT(-1) space).

Theorem (Guéritaud-Kassel, [START_REF] Guéritaud | Maximally stretched laminations on geometrically finite hyperbolic manifolds[END_REF]). Let S be a closed oriented surface of negative Euler characteristic, j a Fuchsian representation of π 1 (S) into PSL(2, R), and ρ a representation of π 1 (S) into the isometry group of a Riemannian CAT(-1) space. Then j strictly dominates ρ if and only if the length spectrum of j strictly dominates the length spectrum of ρ.

Remark 0.3. In [START_REF] Guéritaud | Maximally stretched laminations on geometrically finite hyperbolic manifolds[END_REF], Guéritaud and Kassel study the minimal Lipschitz constant Lip(j, ρ) and its relation with the comparison of the length spectra of j and ρ in a broader context. Namely, they consider j a discrete and faithful representation of the fundamental group of a geometrically finite hyperbolic n-manifold into Isom(H n ). On the other side, they also assume that ρ takes values in the isometry group of H n . However, as François Guéritaud pointed out to us, their proof would work when replacing H n by any Riemannian CAT(-1) space, since it relies on an equivariant Kirzbraun-Valentine theorem than mainly requires the target space to be "more negatively curved" than the base.

Given a representation ρ : π 1 (S) → Isom(M ), it is natural to ask whether ρ can be dominated by a Fuchsian representation and, if so, what the domain of Fuchsian representations dominating ρ looks like. The first question was answered by Deroin and the author in [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF] when (M, g M ) is a Riemannian CAT(-1) space. This applies for instance when M is the symmetric space of a simple Lie group of real rank 1 (with a suitable normalization of the metric), and in particular for representations in PSL(2, R). In that case, it was obtained independently and with other methods by Guéritaud-Kassel-Wolff [START_REF] Guéritaud | Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces[END_REF].

Theorem (Deroin-Tholozan). Let S be a surface of negative Euler characteristic, (M, g M ) a Riemannian CAT(-1) space and ρ a representation of π 1 (S) into Isom(M ). Then either ρ is Fuchsian in restriction to some stable totally geodesic 2-plane of curvature -1 embedded in M , or there exists a Fuchsian representation that strictly dominates ρ.

Remark 0.4. Note that, by a simple volume argument (see [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF], proposition 2.1), a Fuchsian representation cannot be strictly dominated. Thus the theorem is optimal. 0.3. Topology of Adm(S). Let (M, g M ) be a Riemannian CAT(-1) space. Consider a Fuchsian representation j : π 1 (S) → PSL(2, R) and a representation ρ : π 1 (S) → Isom(M ). One easily checks that Lip(j, ρ) only depends on the class of j under conjugation by PSL(2, R) and on the class of ρ under conjugation by Isom(M ). We can thus see Lip as a function defined on T (S) × Rep(S, Isom(M )) , where Rep(S, Isom(M )) denotes the space of conjugacy classes of representations of π 1 (S) into Isom(M ), and T (S) the space of classes of representations of Euler class -χ(S) modulo conjugation, which is well-known to be isomorphic to the Teichmüller space of S (see section 1.2.2). Proposition 0.5. Let M be a Riemannian CAT(-1) space. Then the function

Lip : T (S) × Rep(S, Isom(M )) → R + is continuous.
This proposition is proved by Guéritaud and Kassel in a slightly different context [START_REF] Guéritaud | Maximally stretched laminations on geometrically finite hyperbolic manifolds[END_REF]. For completeness, we include a proof in the appendix of this article.

Let us denote by Dom(ρ) the domain of T (S) consisting of representations of Euler class -χ(S) strictly dominating a given representation ρ, an by Dom(S, Isom(M )) the domain of pairs (j, ρ) ∈ T (S) × Rep(S, Isom(M )) such that j strictly dominates ρ. By continuity of the function Lip, Dom(ρ) and Dom(S, Isom(M )) are open domains. To prove their theorem, Deroin and the author consider in [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF] a certain map Ψ ρ : T (S) → T (S) and show that its image lies in Dom(ρ). The map Ψ ρ is constructed this way: given a point X in T (S) one can associate to the pair (X, ρ) a holomorphic quadratic differential Φ(X, ρ) on X, obtained as the Hopf differential of an equivariant harmonic map (see section 1.2.1). By a theorem of Sampson-Hitchin-Wolf (see section 1.2.2) there is a unique Fuchsian representation j up to conjugacy such that Φ(X, j) = Φ(X, ρ). We set Ψ ρ (X) = j. (More details are given in section 2.1.) Here, we prove that this map is a homeomorphism from T (S) to Dom(ρ) and that it varies continuously with ρ. We thus obtain the following: Theorem 1. Let S be a closed oriented surface of negative Euler characteristic and (M, g M ) a Riemannian CAT(-1) space. Then the domain Dom(S, Isom(M )) is homeomorphic to

T (S) × Rep nf (S, Isom(M )) ,
where Rep nf (S, Isom(M )) denotes the domain of Rep(S, Isom(M )) consisting of representations that are not Fuchsian in restriction to some totally geodesic plane of curvature -1.

What's more, this homeomorphism is fiberwise, meaning that it restricts to a homeomorphism from T (S) × {ρ} to Dom(ρ) × {ρ} for any ρ ∈ Rep nf (S, Isom(M, g M )).

As a consequence of Theorem 1, the connected components of Dom(S, Isom(M )) are in 1-1 correspondance with the connected components of Rep nf (S, Isom(M )). In the case where M is the Poincaré half-plane, we have, by Kassel's theorem: Dom(S, Isom + (H 2 , g P )) = Adm(S) . In particular, it has 4g -5 connected components, classified by the Euler class of the non-Fuchsian representation in each pair. 0.4. Further applications. Note that the map Ψ ρ depends non trivially on the choice of a normalization of the metric on M . Fix a metric g 0 on M of sectional curvature ≤ -1 and a constant α ≥ 1. Then the metric 1 α 2 g 0 still has sectional curvature ≤ -1. To mark the dependence of the function Lip on the metric on the target, we denote by Lip g (j, ρ) the minimal Lipschitz constant of a (j, ρ)-equivariant map from (H 2 , g P ) to (M, g). Then we clearly have Lip 1 α 2 g 0 = 1 α Lip g 0 . Hence, if we apply theorem 1 to (M, 1 α 2 g 0 ), we obtain a description of the space of pairs (j, ρ) ∈ T (S) × Rep(π 1 (S), G) such that Lip g 0 (j, ρ) < α. For α > 1, the curvature of (M, 1 α 2 g 0 ) is eveywhere < -1. Therefore, there is no totally geodesic plane of curvature -1 in (M, 1 α 2 g 0 ) and we obtain the following: Corollary 3. Let S be a closed oriented surface of negative Euler characteristic, (M, g M ) a Riemannian CAT(-1) space and α a constant bigger than 1. Then the domain

(j, ρ) ∈ T (S) × Rep(S, Isom(M )) | Lip g M (j, ρ) < α is fiberwise homeomorphic to T (S) × Rep(S, Isom(M )) .
When applied to M = (H 2 , g P ), we obtain a description of left open balls for Thurston's asymmetric distance on T (S). Recall that this distance, introduced in [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF], is defined by

d T h (j, j ′ ) = log Lip g P (j, j ′ )
for j and j ′ any two representations of Euler class -χ(S). The function d T h is continuous, positive whenever j and j ′ are distinct, and satisfies the triangular inequality. However, it is not symmetric.

Fix a point j 0 in T (S) and a constant C > 0. Then, using the convexity of length functions on T (S) [START_REF] Wolpert | Geodesic length functions and the Nielsen problem[END_REF], one can see that the domain

{j ∈ T (S) | d T h (j 0 , j) < C}
is an open convex domain of T (S) for the Weil-Petersson metric (cf. section 1.2.2). In particular, it is homeomorphic to a ball of dimension -3χ(S). Since the distance is asymmetric, it is not clear whether the same holds for

{j ∈ T (S) | d T h (j, j 0 ) < C} . However, since d T h (j, j 0 ) < C ⇔ Lip(j, j 0 ) < e C ,
we can apply corollary 3 and obtain:

Corollary 4. Let S be a closed oriented surface of negative Euler characteristic and j 0 a point in T (S). Then for any C > 0, the domain

{j ∈ T (S) | d T h (j, j 0 ) < C}
is homeomorphic to T (S). In other words, left open balls for Thurston's asymetric distance on T (S) are contractible. 0.5. Structure of the article and strategy of the proof. The article is organized as follows. In the next section we recall some fundamental results about harmonic maps from a surface. In particular, we recall the Corlette-Labourie theorem of existence of equivariant harmonic maps and the Sampson-Hitchin-Wolf parametrization of the Teichmüller space by means of quadratic differentials. In the second section, we start by using those theorems to construct the map Ψ ρ studied in [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF], and we prove that Ψ ρ is a homeomorphism from T (S) to Dom(ρ). To do so, we make explicit the inverse of the map Ψ ρ . It will appear that reverse images of a point j in T (S) by Ψ ρ are exactly critical points of a certain functional F j,ρ . We will prove that when j is in Dom(ρ), the functional F j,ρ is proper and admits a unique critical point which is a global minimum. Hence the map Ψ ρ is bijective. What's more, the functionals F j,ρ vary continuously with (j, ρ), and so does their unique minimum. This will prove the continuity of Ψ -1 ρ and its continuous dependance in ρ.

We wish to thank Richard Wentworth and Mike Wolf for their interest in this work and their help in understanding some regularity question concerning harmonic maps. We also thank Olivier Guichard and Fanny Kassel for useful remarks on a previous version of this paper.

Representations of surface groups, harmonic maps, and Teichmüller space

In this section, we introduce briefly the tools from the theory of harmonic maps that we will need later. We refer to [START_REF] Daskalopoulos | Harmonic maps and Teichmüller theory[END_REF] for a more thorough survey.

1.1. Existence theorems. Recall that the energy density of a non-negative symmetric 2-form h on a Riemannian manifold (X, g) is the function on X defined as

e g (h) = 1 2 Tr(A(h)) ,
where A(h) is the unique field of endomorphisms of the tangent bundle such that for all x ∈ X and all u, v ∈ T x X,

h x (u, v) = g x (u, A(h) x v) .
If f is a map between two Riemannian manifolds (X, g X ) and (Y, g Y ) then its energy density is the energy density on X of the pull-back metric:

e g X (f ) = e g X (f * g Y ) .
This energy density can be integrated against the volume form Vol g X associated to g X , giving the total energy of f :

E g X (f ) = X e g X (f )Vol g X .
The map f is called harmonic if it is, in some sense, a critical point of the total energy. For instance, harmonic maps from R to a Riemannian manifold are geodesics and harmonic maps from a Riemannian manifold to R are harmonic functions. In general, a map f is harmonic if it satisfies a certain partial differential equation that can be expressed as the vanishing of the tension field. We give and use the precise definition of harmonicity in [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF]. Here we will be satisfied with existence results and some fundamental properties of those maps.

The first existence result is due to Eells and Sampson.

Theorem (Eells-Sampson, [START_REF] James Eells | Harmonic mappings of Riemannian manifolds[END_REF]). Let f : (X, g X ) → (Y, g Y ) be a continuous map between two closed Riemannian manifolds. Assume (Y, g Y ) has non-positive sectional curvature. Then there exists a harmonic map f ′ : (X, g X ) → (Y, g Y ) homotopic to f , which minimizes the energy among all maps homotopic to f . Moreover, if the sectional curvature of Y is negative, then this map is unique, unless the image of f ′ is included in a geodesic of Y .

Eells and Sampson's paper contains a thorough study of the analytic aspects of harmonicity, which allows to extend their existence result in several cases. The one we will be interested in is an equivariant version. Consider (X, g X ) a closed Riemannian manifold, ( X, g X ) its universal cover, (M, g M ) another Riemannian manifold and ρ a representation of π 1 (X) into Isom(M ). A map f : X → M is called ρ-equivariant if for all x ∈ X and all γ ∈ π 1 (X), we have

f (γ • x) = ρ(γ) • f (x).
Given such a map, the symmetric 2-form f * g M on X is invariant under the action of π 1 (X) and thus induces a symmetric 2-form on X. We define the energy density and the total energy of f as the energy density and the total energy of this symmetric 2-form.

The equivariant version of Eells-Sampson's theorem is due to Corlette [START_REF] Corlette | Flat G-bundles with canonical metrics[END_REF] in the specific case where M is a symmetric space of non-compact type, and to Labourie [START_REF] Labourie | Existence d'applications harmoniques tordues à valeurs dans les variétés à courbure négative[END_REF] in the more general case of a complete simply connected Riemannian manifold of non-positive curvature. We state it in the particular case where (M, g M ) is a Riemannian CAT(-1) space. This implies that (M, g M ) is Gromov hyperbolic. Recall that, for such a space, there is a good notion of boundary at infinity, such that every isometry of M extends to a homeomorphism of the boudary (see section A of the appendix).

Theorem (Corlette, Labourie). Let (X, g X ) be a closed Riemannian manifold, X its universal cover, and (M, g M ) a Riemannian CAT(-1)-space. Let ρ be a representation of π 1 (X) into Isom(M ). Assume that ρ does not fix a point in ∂ ∞ M . Then there exists a unique harmonic map from ( X, g X ) to (M, g M ) that is ρ-equivariant. This map minimizes the energy among all such equivariant maps.

1.1.1. Dealing with representations fixing a point at infinity. In the sequel, representations fixing a point in ∂ ∞ M will be called parabolic representations. When M is the symmetric space of some rank 1 Lie group G, those are exactly the representations taking values into some parabolic subgroup of G. For some of those representations, the theorem of Corlette and Labourie does not apply, and we must say a few words in order to be able to include them in the proof later.

The main problem with a parabolic representation ρ is that a sequence of equi-Lipschitz ρ-equivariant maps from X to M may not converge. In the appendix, we deal with this issue in order to prove the continuity of the function Lip(j, • ) at a parabolic representation. With the same ideas and a few notions of harmonic analysis, one can prove the following fact:

Proposition 1.1. Let ρ : π 1 (X) → Isom(M )) be a parabolic representation. Let m ρ be the morphism from π 1 (X) to R such that |m ρ (γ)| = l(ρ(γ))
for all γ ∈ π 1 (X) (see lemma B.1 in the appendix).

Let f n be a sequence of ρ-equivariant maps from X to M whose energy converges to the infimum of the energies of all such equivariant maps. Then the symmetric 2-form

f * n g M converges to f * dx 2 ,
where f is a m ρ -equivariant harmonic function from X to R and dx 2 is the canonical metric on R. (The function f is obtained by integrating the unique harmonic 1-form on X whose periods are given by m ρ , and is unique up to a translation.)

In some sense, this m ρ -equivariant function f is the natural extension of the notion of ρ-equivariant harmonic maps to parabolic representations. In particular, when ρ fixes two points in ∂ ∞ M , it stabilizes a unique geodesic c : R → M ; in that case, ρ-equivariant harmonic maps exist and have the form c • f .

1.2. Harmonic maps from a surface, Hopf differential, and Teichmüller space.

From now on we will restrict to harmonic maps from a Riemann surface. In that case, the energy of a map only depends on the conformal class of the Riemannian metric on the base. For the same reason, harmonicity is invariant under a conformal change of the metric.

1.2.1. Hopf differential. Let S be an oriented surface equipped with a Riemannian metric g. Let (M, g M ) be a Riemannian manifold, and f : S → M a smooth map. The conformal class of g induces a complex structure on S. The symmetric 2-form f * g M can thus be uniquely decomposed into a (1, 1) part, a (2, 0) part and a (0, 2) part. One can check that the (1, 1) part is e g (f )g, and we thus have

f * g M = e g (f )g + Φ f + Φf ,
where Φ f is a quadratic differential (i.e. a section of the square of the canonical bundle of (S, [g]) called the Hopf differential of f . The following proposition is classical.

Proposition 1.2. If the map f is harmonic, then its Hopf differential is holomorphic. The converse is true if M is also a surface.

1.2.2. The Teichmüller space. Let S be a closed oriented surface of negative Euler characteristic. Recall that the Teichmüller space of S, denoted T (S), is the space of complex structures on S compatible with the orientation, where two complex structures are identified if there is a biholomorphism between them isotopic to the identity. By Poincaré-Koebe's uniformization theorem, any complex structure on S admits a unique conformal Riemannian metric of constant curvature -1. Therefore, T (S) can also be seen as the space of hyperbolic metrics on S up to isotopy. Lastly, any hyperbolic metric g on S induces a Fuchsian representation j : π 1 (S) → PSL(2, R), such that (S, g) identifies isometrically with j(π 1 (S))\H 2 (j is called a holonomy representation of the metric g). The representation j is unique up to conjugation, has Euler class -χ(S), and T (S) thus identifies canonically with the space of representations from π 1 (S) to PSL(2, R) of Euler class -χ(S) modulo conjugation. Throughout the paper, a point in T (S) is denoted alternately by the letter X when we think of it as the surface S equipped with a complex structure, or by the letter j when we think of it as a Fuchsian representation.

It is well known that the Teichmüller space is a manifold diffeomorphic R -3χ(S) and that it carries a complex structure. Consider two points X 1 and X 2 in T (S) corresponding to two hyperbolic metrics g 1 and g 2 on S. Then, by Eells-Sampson's theorem, there is a unique harmonic map f g 1 ,g 2 : (S, g 1 ) → (S, g 2 ) homotopic to the identity map. Schoen-Yau's theorem [START_REF] Schoen | On univalent harmonic maps between surfaces[END_REF] (also proved by Sampson [START_REF] Sampson | Some properties and applications of harmonic mappings[END_REF] in that specific case) states that this map is a diffeomorphism.

Of course, the map f g 1 ,g 2 depends on the choice of g 1 and g 2 up to isotopy. Actually, fixing g 1 or g 2 , one can choose the other metric so that the identity map itself is harmonic (by replacing g 2 by f * g 1 ,g 2 g 2 or g 1 by f * g 1 ,g 2 g 1 ). On the other hand, the total energy of f g 1 ,g 2 is invariant under changing one of the metrics by isotopy, and thus gives a well defined functional

E : T (S) × T (S) → R + .
Besides, the Hopf differential of f g 1 ,g 2 is invariant under isotopic changes of g 2 , and if h is a diffeomorphism of S, one has

Φ f h * g 1 ,g 2 = h * Φ fg 1 ,g 2 .
The Hopf differential thus induces a well defined map

Φ : T (S) × T (S) → QDT (S) ,
where QDT (S) denotes the complex bundle of holomorphic quadratic differentials on T (S), that is, QD X T (S) is the space of holomorphic quadratic differentials on X.

It is known since the work of Teichmüller that this bundle identifies with the tangent bundle to T (S). Sampson [START_REF] Sampson | Some properties and applications of harmonic mappings[END_REF] proved that the map Φ is an injective immersion. In particular, if one fixes a point X 0 ∈ T (S), the derivative at X 0 of the map Φ(X 0 , • ) provides a new identification of the tangent space T X 0 T (S) with QD X 0 T (S).

Finally, Wolf [START_REF] Wolf | The Teichmüller theory of harmonic maps[END_REF] proved that the map Φ is a global homeomorphism. This was obtained independently by Hitchin [START_REF] Hitchin | The self-duality equations on a Riemann surface[END_REF], as the first construction of a section to the Hitchin fibration.

Theorem (Sampson, Hitchin, Wolf). The map Φ : T (S) × T (S) → QDT (S) is a homeomorphism.

1.2.3.

The Weil-Petersson metric on T (S). Consider X ∈ T (S), and choose g a hyperbolic metric representing X. Then g naturally induces a Hermitian metric on the line bundle K 2 X . In a local conformal coordinate z, write g = α dzdz for some positive function α. Then a section Φ of K 2 X is locally of the form ϕ dz 2 , and the Hermitian metric on K 2 X induced by g is given by

Φ 2 g = |ϕ| 2 α 2 .
(One can check that this function is independent of the choice of a local conformal coordinate.) This Hermitian metric allows to define a Hermitian norm on the complex vector space QD X T (S) of holomorphic quadratic differentials on X, given by

Φ 2 W P = S Φ 2 g Vol g .
Since QD X T (S) naturally identifies with T X T (S), this defines a Riemannian metric on T (S), called the Weil-Petersson metric. Many interesting properties of the Weil-Petersson metric have been proved by Ahlfors [START_REF] Ahlfors | Curvature properties of Teichmüller's space[END_REF] and Wolpert [START_REF] Wolpert | Geodesic length functions and the Nielsen problem[END_REF], among others. Here we will be satisfied with this simple definition.

1.3. Equivariant harmonic maps and functionals on T (S). The theorem of Corlette and Labourie allows to extend the maps E and Φ to T (S) × Rep(S, Isom(M )). Given a point in X 0 ∈ T (S), represented by a hyperbolic metric g 0 , and a point ρ in Rep(π 1 (S), Isom(M )), one can consider f g 0 ,ρ the unique ρ-equivariant harmonic map from ( S, g 0 ) to (M, g M ). (If ρ is parabolic, f g 0 ,ρ will denote instead a m ρ -equivariant harmonic function, see section 1.1.1.)

The symmetric 2-form f * g 0 ,ρ g M (or f * g 0 ,ρ dx 2 when ρ is parabolic) on S is π 1 (S)-invariant, hence the energy density and the Hopf differential of f g 0 ,ρ are also π 1 (S)-invariant. They thus descend on S. We denote them e g 0 (f g 0 ,ρ ) and Φ fg 0 ,ρ .

The energy density and the Hopf differential of f g 0 ,ρ only depend on the conjugacy class of the representation ρ. We thus obtain two well-defined maps

E : T (S) × Rep(S, Isom(M )) → R + (X 0 , ρ) → S e g 0 (f g 0 ,ρ )Vol 0 and Φ : T (S) × Rep(S, Isom(M )) → QDT (S) (X 0 , ρ) → Φ fg 0 ,ρ . Remark 1.3. If M = H 2 , we have Isom + (M ) ≃ PSL(2, R).
Consider two points X, X 1 ∈ T (S), and let j 1 be a holonomy representation of a hyperbolic metric g 1 corresponding to X 1 . Then a harmonic map from X to X 1 isotopic to the identity lifts to a j 1 -equivariant harmonic map from X to H 2 . We thus have E(X, j 1 ) = E(X, X 1 ) and Φ(X, j 1 ) = Φ(X, X 1 ) . In other words, the new definition of E and Φ extends the one given in the previous paragraph, via the identification of T (S) with the component of maximal Euler class in Rep(S, PSL(2, R)).

Proposition 1.4. The function E(X, ρ) and the map Φ(X, ρ) are continuous with respect to X and ρ.

Proof. This is a classical consequence of the ellipticity of the equation defining harmonic maps. If (j n , ρ n ) converges to (j, ρ), the derivatives of a (j n , ρ n )-equivariant harmonic map f n can be uniformly controlled by its total energy. If ρ is not parabolic, one can deduce that the sequence f n converges in C 1 topology to a (j, ρ)-equivariant harmonic map f (see section C.3 in the appendix). The proposition easily follows.

The case where ρ is parabolic can also be solved using the same ideas as those in the proof of the continuity of the minimal Lipschitz constant.

We will use the following important results. Proposition 1.5 (see [START_REF] Wentworth | Energy of harmonic maps and Gardiner's formula[END_REF]). Let (M, g M ) be a Riemannian CAT(-1) space and ρ a representation of π 1 (S) into Isom(M ). Then the functional

E( • , ρ) : T (S) → R + X → E(X, ρ)
is C 1 and its differential at a point X 0 ∈ T (S) is given by

Grad W P E( • , ρ)(X 0 ) = -Φ(X 0 , ρ)
(where Grad W P denotes the gradient with respect to the Weil-Petersson metric).

Theorem 1.6 (Tromba,[START_REF] Tromba | Teichmüller theory in Riemannian geometry[END_REF]). Consider a point j in T (S). Then the function

E( • , j) : T (S) → R + X → E(X, j)
is proper.

2. The homeomorphism from T (S) to D(ρ)

We are now in possession of all the tools required to define the map Ψ ρ introduced in [7] and to prove that it is a homeomorphism from T (S) to Dom(ρ).

2.1. Construction of the map Ψ ρ . Let S be a closed oriented surface of negative Euler characteristic and ρ a representation of π 1 (S) into the isometry group of a Riemannian CAT(-1) space (M, g M ). Let X 1 be a point in T (S). Then Φ(X 1 , ρ) is a holomorphic quadratic differential on X 1 , and the theorem of Sampson-Hitchin-Wolf asserts that there is a unique point j 2 in T (S) such that Φ(X 1 , j 2 ) = Φ(X 1 , ρ). Setting

Ψ ρ (X 1 ) = j 2 ,
we obtain a well-defined map Ψ ρ : T (S) → T (S) . This map only depends on the class of ρ under conjugation by Isom(M ). We can thus define a map

Ψ : T (S) × Rep(S, Isom(M )) → T (S) × Rep(S, Isom(M )) (X, ρ) → (Ψ ρ (X), ρ)
. In [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF], the following is proved: Theorem (Deroin-Tholozan). Either ρ preserves a totally geodesic 2-plane of curvature -1 in restriction to which it is Fuchsian, or the image of the map Ψ ρ lies in Dom(ρ). In particular, Dom(ρ) is non empty.

2.2.

Surjectivity of the map Ψ ρ . We first prove the following Proposition 2.1. The map Ψ ρ : T (S) → Dom(ρ) is surjective.

Proof. Fix j 0 ∈ T (S) and ρ ∈ Rep(S, Isom(M )). Let us introduce the functional

F j 0 ,ρ : T (S) → R X → E(X, j 0 ) -E(X, ρ) .
By proposition 1.5, the map F j 0 ,ρ is C 1 and its gradient with respect to the Weil-Petersson metric is given by Grad X (F j 0 ,ρ ) = -Φ(X, j 0 ) + Φ(X, ρ) . Hence X 1 is a critical point of F j 0 ,ρ if and only if Φ(X 1 , j 0 ) = Φ(X 1 , ρ), which means precisely that Ψ ρ (X 1 ) = j 0 . Proving that j 0 is in the image of Ψ ρ is thus equivalent to proving that the map F j 0 ,ρ admits a critical point. This will be a consequence of the following lemma: Lemma 2.2. For j 0 ∈ T (S) and ρ ∈ Rep(S, Isom(M )), we have the following inequality:

F j 0 ,ρ ≥ (1 -Lip(j 0 , ρ)) E( • , j 0 ) .
Proof. Let f be a (j 0 , ρ)-equivariant Lipschitz map from H 2 to M with Lipschitz constant Lip(j 0 , ρ) + ε. Let X be a point in T (S) represented by some hyperbolic metric g, and let h be the unique j 0 -equivariant map from X to H 2 . We thus have E g (h) = E(X, j 0 ). Besides, the map f • h is a ρ-equivariant map from X to M , and we thus have

E(X, ρ) ≤ E g (f • h) . Since f is (Lip(j 0 , ρ) + ε)-Lipschitz, we have E(X, ρ) ≤ E g (f • h) ≤ (Lip(j 0 , ρ) + ε)E g (h) = (Lip(j 0 , ρ) + ε)E(X, j 0 ) ,
from which we get

F j 0 ,ρ (X) = E(X, j 0 ) -E(X, ρ) ≥ (1 -Lip(j 0 , ρ) -ε)E(X, j 0 ).
This being true for any ε > 0, we obtain the required inequality. Now, by Theorem 1.6, the map X → E(X, j 0 ) is proper. Therefore, if j 0 is in Dom(ρ), we have (1-Lip(j 0 , ρ)) > 0 and the function F j 0 ,ρ is also proper. Hence F j 0 ,ρ admits a minimum. This minimum is a critical point, and thus a pre-image of j 0 by the map Ψ ρ . We obtain that Ψ ρ : T (S) → Dom(ρ) is surjective.

We proved that if j 0 is in Dom(ρ), the functional F j 0 ,ρ is proper. Note that, conversely, if F j 0 ,ρ is proper, then it admits a critical point. Hence j 0 is in the image of Ψ ρ , which implies that j 0 lies in Dom(ρ) by [START_REF] Deroin | Dominating surface group representations by Fuchsian ones[END_REF]. We thus obtain the following corollary that might be interesting in its own way:

Corollary 2.3. The following are equivalent:

(i) the map F j 0 ,ρ is proper, (ii) the map F j 0 ,ρ admits a critical point, (iii) the representation j 0 strictly dominates ρ.

2.3.

Injectivity of the map Ψ ρ . To prove injectivity, we need to prove that when j 0 is in Dom(ρ), the critical point of F j 0 ,ρ is unique. To do so, we prove that any critical point of F j 0 ,ρ is a strict minimum of F j 0 ,ρ . Let X 1 be a critical point of F j 0 ,ρ , and X 2 another point in T (S). Choose a hyperbolic metric g 1 on S representing X 1 . Let g 0 be the hyperbolic metric of holonomy j 0 such that Id : (S, g 1 ) → (S, g 0 ) is harmonic, and g 2 the hyperbolic metric representing X 2 such that Id : (S, g 2 ) → (S, g 0 ) is harmonic. Let f : ( S, g 1 ) → (M, g M ) be a ρ-equivariant harmonic map. We have the following decompositions:

g 0 = e g 1 (g 0 )g 1 + Φ + Φ, f * g M = e g 1 (f )g 1 + Φ + Φ, g 2 = e g 1 (g 2 )g 1 + Ψ + Ψ
, where Φ and Ψ are quadratic differentials on S, with Φ holomorphic with respect to the complex structure induced by g 1 . Note that the same Φ appears in the decomposition of g 0 and f * g M because X 1 is a critical point of F j 0 ,ρ , and thus Φ(X 1 , j 0 ) = Φ(X 1 , ρ). Lemma 2.4. We have the following identity:

E g 2 (g 0 ) -E g 2 (f * g M ) = S 1 1 - 4 Ψ 2 g 1 eg 1 (g 2 ) 2 (e g 1 (g 0 ) -e g 1 (f * g M )) Vol g 1 .
Proof of lemma 2.4. This is a rather basic computation that we will carry out in local coordinates. Let z = x + iy be a local complex coordinate with respect to which g 1 is conformal. We denote by • , • the standard scalar product on C Any symmetric 2-form on C can be written under the form • , G • , where G is a field of symmetric endomorphisms of R 2 depending on the coordinates (x, y). We will represent such an endomorphism by its matrix in the canonical frame ( ∂ ∂x , ∂ ∂y ). In local coordinates, we can thus write

g 0 = • , G 0 • , g 1 = • , G 1 • , g 2 = • , G 2 • , f * g M = • , G f • .
Now, since g 1 is conformal with respect to the coordinate z, we have g 1 = α • , • for some positive function α, and we can write

Φ = ϕ dz 2 , Ψ = ψ dz 2 ,
for some complex valued functions ϕ and ψ. (Since Φ is holomorphic, ϕ must be holomorphic, but we won't need it for our computation.)

We can now express G 0 , G 1 , G 2 , G f , Vol g 1 and Vol g 2 in terms of α, e g 1 (g 0 ), e g 1 (g 2 ), e g 1 (f ), ϕ and ψ. We easily check that

G 1 = α 0 0 α , G 0 = αe g 1 (g 0 ) + 2Re(ϕ) -2Im(ϕ) -2Im(ϕ) αe g 1 (g 0 ) -2Re(ϕ) , G f = αe g 1 (g f ) + 2Re(ϕ) -2Im(ϕ) -2Im(ϕ) αe g 1 (g f ) -2Re(ϕ) , G 2 = αe g 1 (g 2 ) + 2Re(ψ) -2Im(ψ) -2Im(ψ) αe g 1 (g 2 ) -2Re(ψ) , Vol g 1 = αdzdz , Vol g 2 = det G 2 dzdz = α 2 e g 1 (g 2 ) 2 -4|ψ| 2 dzdz .
We now want to express e g 2 (g 0 ) and e g 2 (f * g M ). To do so, note that we can write

g 0 ( • , • ) = • , G 0 • = • , G 2 (G -1 2 G 0 ) • = g 2 ( • , G -1 2 G 0 • ) .
By definition of the energy density, we thus obtain that

e g 2 (g 0 ) = 1 2 Tr(G -1 2 G 0 ) = 1 2 Tr 1 det G 2 αe g 1 (g 2 ) -2Re(ψ) 2Im(ψ) 2 Im(ψ) αe g 1 (g 2 ) + 2Re(ψ) × αe g 1 (g 0 ) + 2Re(ϕ) -2Im(ϕ) -2 Im(ϕ) αe g 1 (g 0 ) -2Re(ϕ) = 1 α 2 e g 1 (g 2 ) 2 -4|ψ| 2 α 2 e g 1 (g 2 )e g 1 (g 0 ) -(ϕ ψ + φψ) .
Similarly, we get that

e g 2 (f * g M ) = 1 α 2 e g 1 (g 2 ) 2 -4|ψ| 2 α 2 e g 1 (g 2 )e g 1 (f ) -(ϕ ψ + φψ) .
When computing the difference, the terms ϕ ψ + φψ simplify. (Here we use the fact that f * g M and g 0 have the same (2, 0)-part.) We eventually obtain

(e g 2 (g 0 ) -e g 2 (f * g M )) Vol g 2 = α 2 e g 1 (g 2 )(e g 2 (g 0 ) -e g 2 (f * g M )) α 2 e g 1 (g 2 ) 2 -4|ψ| 2 dzdz = e g 2 (g 0 ) -e g 2 (f * g M ) 1 -4 Ψ 2 g 1 eg 1 (g 2 ) 2
Vol g 1 .

Now, the parameters of the last expression are well defined functions on S and the identity is true in any local chart. It is thus true everywhere on S and, when integrating, we obtain lemma 2.4.

From lemma 2.4, we obtain that

E g 2 (g 0 ) -E g 2 (f * g M ) ≥ S (e g 1 (g 0 ) -e g 1 (f * g M ))Vol g 1 = E g 1 (g 0 ) -E g 1 (f * g M ) = F j 0 ,ρ (X 1 ) ,
with equality if and only if Ψ g 1 ≡ 0, that is, if g 1 is conformal to g 2 .

On the other side, we have E g 2 (g 0 ) = E(X 2 , j 0 ) (since, by hypothesis, the identity map from (S, g 2 ) to (S, g 0 ) is harmonic) and E g 2 (f * g M ) ≥ E(X 2 , ρ), from which we deduce that

E g 2 (g 0 ) -E g 2 (f * g M ) ≤ F j 0 ,ρ (X 2 ) .
Combining the two inequalities, we obtain that

F j 0 ,ρ (X 2 ) ≥ F j 0 ,ρ (X 1 ) , with equality if and only if X 1 = X 2 .
Now, if X 1 and X 2 are two critical points of F j 0 ,ρ , then by symmetry we must have F j 0 ,ρ (X 2 ) = F j 0 ,ρ (X 1 ), and therefore X 2 = X 1 . The functional F j 0 ,ρ admits a unique critical point, and j 0 admits a unique pre-image by Ψ ρ . Thus Ψ ρ is injective.

bi-continuity of Ψ.

Recall that one has, by definition,

(X, Ψ ρ (X)) = Φ -1 (X, Φ(X, ρ)) ,
where Φ -1 denotes the inverse of the map Φ : T (S) × T (S) → QDT (S), which is a homeomorphism by Sampson-Hitchin-Wolf's theorem. Therefore, by proposition 1.4, the maps Ψ ρ and Ψ are continuous.

Let us now prove that Ψ -1 is continuous. We saw that Ψ -1 (j, ρ) is the unique critical point of a proper function F j,ρ on T (S) which depends continuously on j and ρ. The continuity of Ψ -1 will follow from the fact that the functions F j,ρ are in some sense locally uniformly proper.

Definition 2.5. Let X and Y be two metric spaces, and (F y ) y∈Y a family of continuous functions from X to R depending continuously on y for the compact-open topology. We say that the family (F y ) y∈Y is uniformly proper if for any C ∈ R, there exists a compact subset K of X such that for all y ∈ Y and all x ∈ X\K, F y (x) > C.

We say that the family (F y ) y∈Y is locally uniformly proper if for any y 0 ∈ Y , there is a neighbourhood U of y 0 such that the sub-family (F y ) y∈U is uniformly proper. Proposition 2.6. Let X and Y be two metric spaces and (F y ) y∈Y a locally uniformly proper family of continuous functions from X to R depending continuously on y (for the compactopen topology). Assume that each F y achieves its minimum at a unique point x m (y) ∈ X. Then the function y → x m (y) is continuous.

Proof. Let us denote by m(y) = F y (x m (y)) the minimum value of F y . Fix y 0 ∈ Y . Let U be a neighbourhood of y 0 and K a compact subset of X such that for all y ∈ U and all x ∈ X\K, we have

F y (x) > m(y 0 ) + 1 . For ε > 0, define V ε = {x ∈ X | F y 0 (x) < m(y 0 ) + ε} .
Since F y 0 is proper and achieves its minimum at a single point x m (y 0 ), the family (V ε ) ε>0 forms a basis of neighbourhoods of x m (y 0 ). Let U ε be a neighbourhood of y 0 included in U such that for all y ∈ U ε and all x ∈ K,

|F y (x) -F y 0 (x)| < ε 2 .
(U ε exists because the map y → F y is continuous for the compact-open topology.) Since x m (y 0 ) is obviously in K, we have for all y ∈ U ε ,

F y (x m (y 0 )) < m(y 0 ) + ε 2 ,
hence the minimum value m(y) of F y is smaller than m(y 0 ) + ε 2 . In particular, for ε < 2, this minimum is achieved in K (since outside K, we have F y ≥ m(y 0 ) + 1). We thus have x m (y) ∈ K, from which we deduce

F y 0 (x m (y)) < F y (x m (y)) + ε 2 = m(y) + ε 2 < m(y 0 ) + ε .
We have thus proved that x m (y) ∈ V ε for all y ∈ U ε . Since (V ε ) ε>0 is a basis of neighbourhoods of x m (y 0 ), this proves that y → x m (y) is continuous at y 0 .

To prove the continuity of Ψ -1 , we can apply proposition 2.6 to the family F j,ρ of functions on T (S) depending on the parameter (j, ρ) ∈ Dom(S, Isom(M )). The continuity of (j, ρ) → F j,ρ comes from proposition 1.4. The only thing we need to check is thus that the family (F j,ρ ) (j,ρ)∈Dom(S,Isom(M ))

is locally uniformly proper. But this follows easily from the continuity of the minimal Lipschitz constant. Indeed, let (j 0 , ρ 0 ) be a point in Dom(S, Isom(M )). We thus have Lip(j 0 , ρ 0 ) < 1. By continuity of the function Lip, there exists a neighbourhood U of (j 0 , ρ 0 ) and a λ < 1 such that for all (j, ρ) ∈ U , we have Lip(j, ρ) ≤ λ. By lemma 2.2, we thus have

F j,ρ (Y ) ≥ 1 -λ 2 E(Y, j)
for all (j, ρ) ∈ U and all Y ∈ T (S). Since the function Y → E(Y, j) is proper by theorem 1.6, we obtain that the family (F j,ρ ) is uniformly proper on U . Hence it is locally uniformly proper. By proposition 2.6, we deduce that the unique minimum of F j,ρ varies continuously with (j, ρ). Since this minimum is precisely Ψ -1 (j, ρ), we proved that Ψ -1 is continuous. This concludes the proof of theorem 1.

Proposition A.2. Hyperbolic angles satisfy a triangular inequality: if x, y, z and t are four points in M , then ∠ hyp (y, x, t) ≤ ∠ hyp (y, x, z) + ∠ hyp (z, x, t) .

Proposition A.3. Let τ n be a sequence of triangles in M such that the lengths of the edges of τ n go to +∞ and such that the hyperbolic angles at the vertices converge. Then at least two of these angles go to 0.

Boundary at infinity. A CAT(-1) space M is in particular hyperbolic in the sense of Gromov. One can thus define the boundary at infinity of M as the quotient of the space of geodesic rays going at speed 1 by the equivalence relation that identifies two rays γ 1 and γ 2 if and only if the distance between γ 1 (t) and γ 2 (t) is bounded when t goes to +∞. This boundary at infinity, denoted ∂ ∞ M , gives a compactification of M . When M is a Riemannian CAT(-1) space, M ∪ ∂ ∞ M is homeomorphic to a closed ball. Every geodesic γ : R → M converges to a point γ + in ∂ ∞ M when t goes to +∞ and a point γ -when t goes to -∞.

Conversely, any two points in ∂ ∞ M are the extremities of a unique geodesic in M . The hyperbolic angles extend to the boundary, meaning that if x is a point in M and (y n ) and (z n ) two sequences in M converging respectively to p and q in ∂ ∞ M , then ∠ hyp (y n , x, z n ) converges to a value that only depends on x, p and q. We will denote it ∠ ∞ (p, x, q). The function

d x : ∂ ∞ M × ∂ ∞ M → R + (p, q) → ∠ ∞ (p, x, q) is a distance on ∂ ∞ M .
Busemann functions. Fix a point p in ∂ ∞ M and let x 0 and x be two points in M . Consider the geodesic ray γ : R + → M starting at x and pointing toward p. Then d(x 0 , γ(t)) -t converges to a number denoted B p,x 0 (x). By triangular inequality, we clearly have [START_REF] Ahlfors | Curvature properties of Teichmüller's space[END_REF] |B p,x 0 (x)| ≤ d(x 0 , x) .

The function B p,x 0 is called a Busemann function. This function only depends on x 0 up to a translation. Indeed, given x 0 , x 1 and x in M , we have

(2) B p,x 0 (x) = B p,x 0 (x 1 ) + B p,x 1 (x) .

The level sets of the Busemann function B p,x 0 are called horospheres tangent to p. When M is a Riemannian CAT(-1) space of dimension n, horospheres are smooth hypersurfaces diffeomorphic to R n-1 . They form a foliation orthogonal to the 1-dimensional foliation consisting of geodesics pointing toward p.

For every t ∈ R + , we will denote by F p,t the map sending a point x in M to the point γ x,p (t), where γ x,p is the geodesic ray starting at x and pointing toward p. The map F p,t preserves the foliation of M by horospheres tangent to p and induces a surjection from B -1 p,x 0 (a) to B -1 p,x 0 (a+t) for any a ∈ R (figure 1). When M is Riemannian, this surjection is a diffeomorphism.

Proposition A.4. The function F p,t is 1-Lipschitz and satisfies the following inequality:

|B p,x (y)| ≤ d (F p,t (x), F p,t (y)) ≤ |B p,x (y)| + e -t d(x, y) .
In particular, the distance between F p,t (x) and F p,t (y) converges to |B p,x (y)| when t goes to +∞. p,x 0 (a) to the horosphere B -1 p,x 0 (a + t).

Isometries. Isometries of M extend to homeomorphisms of M ∪ ∂ ∞ M . They can have three different types of dynamics, depending on their translation length. Let g be an isometry of M . If l(g) = 0, then g fixes either a point in M -in that case g is called elliptic -or a single point in ∂ ∞ M -it is then called parabolic. An elliptic isometry preseves the distance to a fixed point and has bounded orbits, while a parabolic isometry preserves the horospheres tangent to its fixed point.

When l(g) > 0, the isometry g is called hyperbolic. In that case, the set of points x in M for which d(x, g • x) = l(g) forms a geodesic in M called the axis of g. This axis is preserved by g, and g acts along its axis as a translation of length l(g). The extremities of the axis in ∂ ∞ M are fixed by g. Proposition A.5. Let g be an isometry of M fixing a point p in ∂ ∞ M . Then the number

B p,x (g • x)
does not depend on x and verifies

|B p,x (g • x)| = l(g) .
Properties of hyperbolic isometries. Let us finally state a few properties of hyperbolic isometries that will be useful in the next section. The first one asserts that elements of M that "do not move too much" under the action of a hyperbolic isometry g must be close to the axis of g. Proposition A.6. For any ε > 0 and any constant k, there exists a constant C(ε, k) such that if g is an isometry of M such that l(g) ≥ ε and x a point in M such that According to equation ( 2), for any g, h in Γ, we have

m(gh) = B p,x 0 (gh • x 0 ) = B p,x 0 (h • x 0 ) + B p,h•x 0 (gh • x 0 ) .
By proposition A.5, we get

B p,h•x 0 (gh • x 0 ) = B p,x 0 (g • x 0 ) hence m(gh) = B p,x 0 (g • x 0 ) + B p,x 0 (h • x 0 ) = m(g) + m(h) .
The function m is thus a morphism from Γ to R.

(ii) ⇒ (i). In order to prove the converse, we have to consider two distinct cases.

• Case 1: m ≡ 0. In that case, Γ contains hyperbolic elements. Let us prove that all the axes of those elements have a common extremity. It is then easy to deduce that this common extremity is fixed by Γ.

Assume that there exist g and h two hyperbolic isometries in Γ whose axes have all extremities disjoint. Up to replacing g and h by some powers, we can assume that the group generated by g and h has a Schottky-type dynamics (figure 2).The general idea is then that the orbit of x under this group must be quasi-isometric to a free group with two generators equipped with its Cayley metric, which contradicts the fact that the translation lengths of the elements of this group are given by a morphism with values in R.

To be more precise, we can apply proposition A.8 to g and h and consider (U sends the complement of

+ n ), (U - n ), (V + k ) and (V - k ) four sequences of compact subsets in M ∪ ∂ ∞ M such that U + n = g + , U - n = g -, V + k = h + and V - k = h -, and such that g n sends the complement of U - n into U + n , g -n
U + n into U - n , h k sends the complement of V - k into V + k and h -k sends the complement of V + k into V - k .
For n and k large enough, the sets U + n , U - n , V + k and V - k are pairwise disjoint and the isometry g n h k sends the complement of V - k into U + n . Since U + n = g + and V - k = h -, we obtain from proposition A.9 that the translation length of g n h k goes to +∞ when n and k go simulatenously to +∞. Replacing h by h -1 , one would obtain that the translation length of g n h -k goes to infinity when n and k go to +∞.

On the other side, we have

l(g n h k ) = |n m(g) + k m(h)| ,
and since m(g) and m(h) are non-zero, one can find, for any integer n, an integer k such that

l(g n h k ) ≤ |m(h)| .
This contradicts the fact that l(g n h k ) goes to infinity when n and k go to infinity. This proves that the axes of any two hyperbolic elements in Γ have a common extremity. One can easily deduce from this fact that all those axes have a common extremity which is fixed by Γ.

• Case 2: m ≡ 0. This is the very specific case where no element of Γ is hyperbolic. Let x be a point in M . If the orbit of x under Γ is bounded, then Γ stabilizes this orbit and fixes the center of the smallest closed ball containing this orbit. Otherwise, let us shaw that the closure of Γ • x contains a unique point in ∂ ∞ M . This point will therefore be fixed by Γ. The argument that we use is a variation on a classical argument in the study of Gromov-hyperbolic spaces (see for instance [START_REF] Ghys | Sur les groupes hyperboliques d'apres Mikhael Gromov[END_REF]).

Let us assume by contradiction that the closure of Γ • x contains two distinct points p and p ′ of ∂ ∞ M . Let (g n ) and (h n ) be two sequences of elements in Γ such that

g n • x -→ n→+∞ p Figure 3. The isometry h -1 n g n is hyperbolic. and h n • x -→ n→+∞ p ′ .
Since p and p ′ are distinct, the angle

α n = ∠ hyp (g n • x, x, h n • x)
converges to α = ∠ ∞ (p, x, p ′ ) which is non-zero. Since g n is not hyperbolic and since d(x, g n • x) goes to infinity, we obtain from proposition A.10 that the angle ∠ hyp (g n • x, x, g -1 n • x) goes to 0 when n goes to infinity.

What's more, the lengths of the triangle τ (x, h n •x, g n •x) all go to +∞ and, from proposition A.3, we deduce that the angle ∠ hyp (x, h n • x, g n • x) goes to 0 when n goes to +∞. Applying the isometry g -1 n , we get that the angle ∠ hyp (g -1 n • x, x, g -1 n h n • x) also goes to 0. By triangular inequality, we thus know that ∠ hyp (g -1 n h n • x, x, g n • x) goes to 0 (the figure 3 helps visualizing those angles). Reversing the roles of g n and h n , one would prove similarly that ∠ hyp (h -1 n g n • x, x, h n • x) goes to 0. We eventually obtain that the angle

∠ hyp (g -1 n h n • x, x, h -1 n g n • x) goes to α = 0. Since the distance d(x, h -1 n g n • x
) grows to infinity, we deduce from proposition A.10 that h -1 n g n is a hyperbolic isometry for n large enough, which contradicts the hypothesis. Therefore, the closure of Γ • x contains a unique point in ∂ ∞ M which is fixed by Γ. This concludes the proof of lemma B.1.

Proof of lemma B.2. Let j : π 1 (S) → Isom(H 2 ) be a Fuchsian representation and ρ : π 1 (S) → Isom(M ) a parabolic representation. According to lemma B.1, there exists a morphism m : π 1 (S) → R such that for all g ∈ π 1 (S),

l(g) = |m(g)| .
Let us prove that Lip(j, ρ) = Lip(j, m) .

Let p ∈ ∂ ∞ M be the fixed point of ρ. Recall that the morphism m is given by

m(γ) = B p,x (ρ(γ) • x) ,
for any point x ∈ M . Consider first a (j, ρ)-equivariant and C-Lipschitz map f : H 2 → M . Fix a base point x 0 ∈ M and set f :

H 2 → R x → B p,x 0 (f (x)) .
We easily check that f is (j, m)-equivariant. Moreover, we have, for any x, y ∈ H 2 ,

| f (y) -f (x)| = |B p,x 0 (f (y)) -B p,x 0 (f (x))| = |B p,f (x) (f (y))| ≤ d(f (x), f (y)) ,
hence f is also C-Lipschitz. We obtain that Lip(j, m) ≤ Lip(j, ρ) .

Let us now prove the converse inequality. Let f be a (j, m)-equivariant and C-Lipschitz function from H 2 to R. Let S × ρ M denote the flat M -bundle

H 2 × M/(j, ρ)(π 1 (S)) , and S × m R the flat R-bundle H 2 × R/(j, m)(π 1 (S)) .
The map from H 2 × M to H 2 × R sending (x, y) to (x, B p,x 0 (y)) induces a fibration

π : S × ρ M → S × m R ,
whose fibers are horospheres tangent to p. Since those horospheres are contractible, π admits a smooth section that allows to lift a (j, m)-equivariant map f :

H 2 → R to a (j, ρ)-equivariant map f : H 2 → M such that f (x) = B p,x 0 (f (x))
for all x ∈ H 2 . This map f is C ′ -Lipschitz for some constant C ′ , but C ′ is a priori greater than C. However, since p is fixed by ρ, the map F p,t commutes with ρ(γ) for all γ ∈ π 1 (S). Consequently, the map F p,t • f is another (j, ρ)-equivariant map. Now, by proposition A.4, for all x and y in H The map F p,t • f is thus (C + C ′ e -t )-Lipschitz. Since this is true for all C > Lip(j, m) and all t ∈ R + , we obtain Lip(j, ρ) ≤ Lip(j, m) .

This concludes the proof of lemma B.2.

Appendix C. Continuity of the minimal Lipschitz constant

We are now able to prove the continuity of the minimal Lipschitz constant. The arguments are not new and the proof essentially follows the one of Guéritaud-Kassel [START_REF] Guéritaud | Maximally stretched laminations on geometrically finite hyperbolic manifolds[END_REF] (where they assume that the target space is the hyperbolic space). As we said before, the tricky part will be to prove lower semi-continuity at a parabolic representation ρ.

C.1. Continuity with respect to j. Let us start by fixing a representation ρ ∈ Rep(S, Isom(M )). The continuity of the map j → Lip(j, ρ) easily follows from the continuity of Thurston's asymmetric distance on T (S) [START_REF] Thurston | Minimal stretch maps between hyperbolic surfaces[END_REF]. Let j 0 : π 1 (S) → Isom(H 2 ) be a Fuchsian representation and fix ε > 0. By continuity of Thurston asymetric distance, there exists a neighbourhood U ε of j 0 such that for all j ∈ U ε , we have Lip(j 0 , j) ≤ 1 + ε and Lip(j, j 0 ) ≤ 1 + ε. Let j be a point in U ε and h : H 2 → H 2 a (j, j 0 )-equivariant and (1 + 2ε)-Lipschitz map. Let otherwise f : H 2 → M be a (j 0 , ρ)-equivariant and (Lip(j 0 , ρ) + ε)-Lipschitz map. Then the map f • h : H 2 → M is (j, ρ)-equivariant and (1 + ε) (Lip(X, ρ) + ε)-Lipschitz. We thus have Lip(j, ρ) ≤ (1 + ε) (Lip(j 0 , ρ) + ε) .

Switching j 0 and j, we get in the same manner that Lip(j 0 , ρ) ≤ (1 + ε) (Lip(j, ρ) + ε) .

We eventually obtain Lip(j 0 , ρ) -Cste ε ≤ Lip(j, ρ) ≤ Lip(j 0 , ρ) + Cste ε for all j ∈ U ε , which proves the continuity of j → Lip(j, ρ) at j 0 . C.2. Upper semi-continuity in ρ. Since the group of isometries of a Riemannian manifold is a Lie group, the space Hom(S, Isom(M )) possesses a structure of real algebraic variety. In particular it is arcwise connected and we just have to show that if (ρ t ) t∈[0,ε[ is a continuous family of representations of π 1 (S) into Isom(M ), then lim sup t→0 Lip(j, ρ t ) ≤ Lip(j, ρ 0 ) .

To this purpose, one can see ρ t as the monodromy of a flat connection ∇ t on a fixed Mbundle E over S ≃ j(π 1 (S))\H 2 , varying continuously with t. Fix C > Lip(j, ρ 0 ) and let f 0 be a (j, ρ 0 )-equivariant and C-Lipschitz map from H 2 to M . The map f 0 induces a section s 0 of the flat bundle (E, ∇ 0 ), which in turn can be seen as a section s t of (E, ∇ t ) associated to a (j, ρ t )-equivariant map f t : H 2 → M . The continuity of t → ∇ t and the compacity of S ensure that f t is (C + α(t))-Lipschitz for some function α(t) going to 0 when t goes to 0. We deduce that lim sup t→0 Lip(j, ρ t ) ≤ C

For any C > Lip(j, ρ 0 ) and thus that lim sup t→0 Lip(j, ρ t ) ≤ Lip(j, ρ 0 ) .

This proves upper semi-continuity.

C.3. Lower semi-continuity in ρ. To prove lower semi-continuity, consider a sequence (ρ n ) of representations into Isom(M ) converging to ρ. Suppose that there exists a sequence f n of (j, ρ n )-equivariant and C-Lipschitz maps. Let us prove that Lip(j, ρ) ≤ C .

Fix a base point x in H 2 . If f n (x) does not escape to infinity in M , then, by Ascoli's theorem, a sub-sequence of (f n ) converges uniformly on every compact set to a C-Lipschitz and ρequivariant map. We conclude that Lip(j, ρ) ≤ C. If f n (x) escapes to infinity, we can assume that f n (x) converges to a point p in ∂ ∞ M .

Proposition C.1. Assume f n (x) converges to p ∈ ∂ ∞ M . Then p is fixed by ρ(γ) for all γ ∈ π 1 (S).

Proof. Fix γ ∈ π 1 (S). Note first that the sequence

d(f n (x), ρ n (γ) • f n (x))
is bounded. Indeed,

d(f n (x), ρ n (γ) • f n (x)) = d(f n (x), f n (j(γ) • x)) ≤ C d(x, j(γ) • x) .
Since p ∈ ∂ ∞ M and since f n (x) converges to p, we can conclude that ρ n (γ) • f n (x) also converges to p. But ρ n (γ) converges uniformly to ρ(γ) on M ∪ ∂ ∞ M . We thus have ρ(γ) • p = p .

We are now left with the case where ρ fixes a point p in ∂ ∞ M . According to lemma B.1, there is a morphism m : π 1 (S) → R such that for all γ ∈ π 1 (S), l(ρ(γ)) = |m(γ)| .

Assume first that m ≡ 0. Then, according to lemma B.2, we have Lip(j, ρ) = Lip(j, m) = 0 and lower semi-continuity is trivial at ρ. Otherwise, let us prove that we can find a sequence (g n ) of isometries of M such that g n • f n (x) is bounded. (Note that this is immediate when M is homogeneous.) Let γ ∈ π 1 (S) be such that ρ(γ) is hyperbolic. Let A be the axis of ρ(γ) and l = l(ρ(γ)). For n big enough, ρ n (γ) is an isometry of axis A n and translation length l n , and we have We saw (propostion C.1) that d (f n (x), ρ n (γ) • f n (x)) remains bounded when n goes to +∞. From A.6, there exists a constant B ′ such that for n big enough,

d(f n (x), A n ) ≤ B ′ .

  Denoting by Rep k (S, PSL(2, R)) the connected component of representations with Euler class k in Rep(S, PSL(2, R)), we obtain: Corollary 2. Let S be a closed oriented surface of negative Euler characteristic. Then the space Adm(S) is homeomorphic to T (S) × χ(S)<k<-χ(S) Rep k (S, PSL(2, R)) .

Figure 1 .

 1 Figure 1. The map F p,t sending the horosphere B -1p,x 0 (a) to the horosphere B -1 p,x 0 (a + t).

  d(x, g • x) ≤ k , then d(x, axis(g)) ≤ C(ε, k) .

(

  ii) there exists a morphism m from Γ to R such that for all g ∈ Γ,l(g) = |m(g)| .Lemma B.2. Let j be a Fuchsian representation from π 1 (S) to Isom(H 2 ) and ρ : π 1 (S) → Isom(M ) a parabolic representation. Let m : π 1 (S) → R be a morphism such thatl(ρ(γ)) = |m(γ)|for all γ ∈ π 1 (S). Denote by Lip(j, m) the minimal Lipschitz constant of a (j, m)-equivariant map from H 2 to R. Then Lip(j, ρ) = Lip(j, m) .Proof of lemma B.1.(i) ⇒ (ii). If Γ has a fixed point in M , then l(g) = 0 for any g ∈ Γ and m is the trivial morphism. Otherwise, let p ∈ ∂ ∞ M be a fixed point of Γ. Choose a base point x 0 in M and set m(g) = B p,x 0 (g • x 0 ) for all g ∈ Γ. According to proposition A.5, we have |m(g)| = l(g) .

Figure 2 .

 2 Figure 2. Schottky-type dynamics.

2

 2 we haved (F p,t • f (x), F p,t • f (y)) ≤ |B p,f (x) (f (y))| + e -t d(f (x), f (y)) ≤ | f (y) -f (x)| + e -t C ′ d(x, y) ≤ C + C ′ e -t d(x, y) .

Appendix

This appendix is devoted to the proof of the following proposition (cf. proposition 0.5):

Proposition. Let S be a closed oriented surface of negative Euler characteristic and M a Riemannian CAT(-1) space. Then the function Lip : T (S) × Rep(S, Isom(M )) → R + (j, ρ) → inf λ | ∃f : H 2 → M (j, ρ)-equivariant and λ-Lipschitz is continuous.

The proof will proceed in three steps. First, we will use the continuity of Thurston's asymmetric distance on T (S) to show that Lip(j, ρ) is continuous in j. Then we will prove upper semi-continuity in ρ with a general argument that allows to deform a (j, ρ)-equivariant map into a (j, ρ ′ )-equivariant map for ρ ′ close to ρ, without increasing too much the Lipschitz constant. Finally, we will prove lower semi-continuity by showing that a sequence of equi-Lipschitz and (j, ρ n )-equivariant maps converges to a (j, ρ)-equivariant map when ρ n converges to ρ as soon as ρ is not parabolic. The tricky part of the proof will be to deal with lower semicontinuity at parabolic representations. This will require some preliminary work on isometry groups of CAT(-1)-spaces.

The first section of this appendix will therefore be a miscellaneous recall of classical facts about CAT(-1) spaces and their isometries. In the second section, we will characterize parabolic representation by the additivity of their length spectrum. Finally, we will prove proposition 0.5 in the third section.

Appendix A. CAT(-1)-spaces and their isometries

Recall that a length space is a metric space in which the distance between two points is defined as the infimum of the lengths of paths joining these two points. If x 1 , x 2 and x 3 are three points of a length space, a triangle with vertices x 1 , x 2 and x 3 (denoted τ (x 1 , x 2 , x 3 )) is the union of three geodesic segments joining x 1 with x 2 , x 2 with x 3 and x 3 with x 1 .

Let (M, g M ) be a complete simply connected Riemannian manifold of sectional curvature bounded above by -1. Then M is in particular a CAT(-1) space, in the following sense:

) is a triangle in H 2 whose edges have the same length as those of τ (x 1 , x 2 , x 3 ), then the map from τ (x 1 , x2 , x3 ) to M that sends xi to x i and that is geodesic on each edge is 1-Lipschitz.

Lots of properties of the hyperbolic plane can be extended to any CAT(-1) space by simple triangle comparison. We did not find a reference specific to CAT(-1) geometry, but the facts we state here are usually particular cases of more general properties of Gromov-hyperbolic spaces or CAT(0) spaces. One can consult [START_REF] Ghys | Sur les groupes hyperboliques d'apres Mikhael Gromov[END_REF] or [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF] for more information.

Hyperbolic angles. Let M be a CAT(-1) space and x, y, z three points in M . We will denote by ∠ hyp (y, x, z) the angle at the vertex x of a hyperbolic triangle τ (x, ȳ, z) whose edges have the same length as those of τ (x, y, z).

Remark A.7. Though we don't need it, we could give an explicit value of C(ε, k) by comparing with the case of H 2 .

The next proposition states that hyperbolic isometries have "north-south" dynamics in the following sense: Proposition A.8. Let g be a hyperbolic isometry of M and g -, g + the extremities of its axis (oriented in the direction in which g translates). Then there exist sequences of compact subsets (U - n ) and

and such that g n sends the complement of U - n into U + n . What's more, this type of dynamics caracterizes hyperbolic isometries: Proposition A.9. Let g n be a sequence of isometries in M . Assume that there exists two distinct points p -and p + in ∂ ∞ M and two sequences of compact subsets (U - n ) and

and such that g n sends the complement of U - n into U + n . Then the translation length of g n goes to +∞ when n goes to +∞. In particular, g n is hyperbolic for n big enough.

Let us finish with a proposition that characterizes hyperbolic isometries by looking only at three points in an orbit.

Proposition A.10. For any ε > 0, there is a constant C such that for any g in Isom(M ), if there exists a point x such that

Again, we could specify the constant C by comparing with the case of H 2 .

Appendix B. Parabolic representations

Recall that a representation ρ : π 1 (S) → Isom(M ) is called parabolic if it fixes a point in ∂ ∞ M . The purpose of this section is to prove two lemmas that will allow us to show lower semi-continuity of Lip(j, • ) at a parabolic representation in the next section. The first lemma characterises parabolic representations by the additivity of their length spectrum, while the second one relates the minimal Lipschitz constant to this length spectrum.

Lemma B.1. Let Γ be a subgroup of Isom(M ). The following are equivalent:

Let a n ∈ A n be such that (a n ) converges to a point a ∈ A. Let y n be the orthogonal projection of f n (x) onto A n . Since ρ n (γ) acts as a translation of length l n along A n , there is an integer

We eventually obtain

and since d M (a n , a) and l n are bounded, ρ n (γ) kn • f n (x) is bounded. Now the map ρ n (γ) kn •f n is (j, ρ n (γ) -kn •ρ)-equivariant and C-Lipschitz (where ρ n (γ) -kn •ρ denotes the composition of ρ with the conjugation by ρ n (γ) -kn ). By Ascoli's theorem, a subsequence of ρ n (γ) kn • f n converges to f ′ , which is C-Lipschitz and (j, ρ ′ )-equivariant for some representation ρ ′ . We thus have Lip(j, ρ ′ ) ≤ C. On the other side, for all γ ′ ∈ π 1 (S), we have l(ρ ′ (γ ′ )) = lim We thus obtain Lip(j, ρ) ≤ C .

This concludes the proof of the lower semi-continuity of the map ρ → Lip(j, ρ).


Juliette Chabassier

Sébastien Imperiale

Fourth order energy-preserving locally implicit discretization for linear wave equations

 provided that some limitations are overcame in the future : treatment of dissipative terms, non trivial boundary conditions, coupling with a PML region, fluid structure coupling . . . Figure 1: Marmousi 2D benchmark domain presenting high velocity contrasts and refined geometry.