Articles

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(avec Vincent Minerbe) A Kummer construction for gravitational instantons [pdf], arXiv:1005.5133 [math.DG].

Abstract. We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.



(avec Marc Herzlich) Analyse sur un demi-espace hyperbolique et polyhomogénéité [pdf], arXiv:1002.4106 [math.DG].

Abstract. We prove local polyhomogeneity of asymptotically real or complex hyperbolic Einstein metrics, with application to unique continuation problems.



(avec Stuart Armstrong) Einstein metrics with anisotropic boundary behavior [pdf], Int. Math. Res. Notices (2010).

Abstract. We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.



Extended correspondence of Kostant-Sekiguchi-Vergne [pdf].

Abstract. The Kostant-Sekiguchi-Vergne is extended from nilpotent orbits to general orbits and their degenerations (e.g. cotangent spaces of real flag manifolds). This note was written for a project with Oscar García-Prada and Ignasi Mundet I Riera, but it will not be published, as I was not aware that most results were already proved by Roger Bielawski: Lie groups, Nahm's equations and hyper-Kähler manifolds, in: Tschinkel, Yuri (ed.), Algebraic groups. Proceedings of the summer school, Göttingen, June 27-July 13, 2005. Göttingen: Universitätsverlag Göttingen. Universitätsdrucke Göttingen. Seminare Mathematisches Institut, 1-17 (2007).



Continuation unique à partir de l'infini conforme pour les métriques d'Einstein [pdf], Math. Res. Lett. 15(6), 1091-1099 (2008).

Abstract. We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.



(avec Rafe Mazzeo) A nonlinear Poisson transform for Einstein metrics on product spaces [pdf], math.DG/0701868. J. Eur. Math. Soc. (à paraître).

Abstract. We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If M is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of M.



(avec Yann Rollin) Wormholes in ACH Einstein manifolds. [pdf], Trans. Amer. Math. Soc. 361(4), 2021-2046 (2009).

Abstract. We give a new construction of Einstein manifolds which are asymptotically complex hyperbolic, inspired by the work of Mazzeo-Pacard in the real hyperbolic case. The idea is to develop a gluing theorem for 1-handle surgery at infinity, which generalizes the Klein construction for the complex hyperbolic metric.



Sur les variétés CR de dimension 3 et les twisteurs. [pdf], Ann. Inst. Fourier 57(4), 1161-1180 (2007).

Abstract. We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.



Asymptotically symmetric Einstein metrics. Transl. from the French by Stephen S. Wilson. (English) SMF/AMS Texts and Monographs 13. American Mathematical Society (AMS) v, 105 p. (2006).



(avec Marc Herzlich, Michel Rumin) Diabatic Limit, Eta Invariants and Cauchy-Riemann Manifolds of Dimension 3. [pdf], Ann. scient. Éc. Norm. Sup. 40(4), 589-631 (2007).

Abstract. We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the eta-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the problem of filling a CR manifold by a complex hyperbolic manifold, and more generally by a Kähler-Einstein or an Einstein metric.



(avec Marc Herzlich) An invariant of Cauchy-Riemann Seifert 3-manifolds and applications. [ps,pdf], math.DG/0407179.

Abstract. (Obsolete article, see next one) We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3-manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a Kähler-Einstein or an Einstein metric.



Autodual Einstein versus Kähler-Einstein. [ps,pdf]. Geom. Funct. Anal. 15(3), 598-633 (2005).

Abstract. Any pseudoconvex domain in C² carries a complete Kähler-Einstein, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere subject to a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kähler-Einstein metrics).



Sur des variétés de Cauchy-Riemann dont la forme de Levi a une valeur propre positive. [ps,pdf]. Math. Z. 249(2), 411-425 (2005).

Abstract. For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.



(avec Marc Herzlich) A Burns-Epstein invariant for ACHE 4-manifolds. [ps,pdf]. Duke Math. J. 126(1), 53-100 (2005).

Abstract. We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kähler-Einstein case. The invariant we obtain is related to the topological number euler-3signature and to contributions depending only on the CR structure at infinity.


(avec Philip Boalch) Wild nonabelian Hodge theory on curves [ps, pdf]. Compositio Math. 140(1), 179-207 (2004).

Abstract. On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order.
The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that they carry hyperKähler metrics, which are complete when the residue of the connection is semisimple.


Métriques autoduales sur la boule [ps,pdf]. Inventiones Math. 148(3), 545-607 (2002).

Abstract. A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball.
When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun.
The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain.
Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-Kähler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m-1)-sphere is the boundary of a quaternionic-Kähler metric on the (4m)-ball.


(avec Marcos Jardim) Asymptotic behaviour and the moduli space of doubly-periodic instantons [ps, pdf]. J.Eur. Math. Soc. 3(4), 335-375 (2001).

Abstract. We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to TxCP¹. The converse statement is also true, namely a holomorphic bundle on TxCP¹ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus.


Métriques d'Einstein asymptotiquement symétriques. Astérisque 265 (2000), 115 pages.

Abstract. In this article, I study asymptotically symmetric Einstein metrics: asymptotically symmetric means that the curvature at infinity is asymptotic to the curvature of a rank one symmetric space of noncompact type (that is, a hyperbolic space). Two constructions of such metrics are given. The first one relies on analysis to prove that the Einstein deformations of complex, quaternionic or octonionic symmetric spaces are in 1-1 correspondence with some Carnot-Carathéodory metrics on the boundary at infinity. In the quaternionic or octonionic cases, I get new objects at infinity which I call quaternionic (or octonionic) contact structures. The second construction is twistorial: given a real analytic quaternionic contact structure, I prove that it is the boundary at infinity of a unique quaternionic-Kähler (and therefore Einstein), asymptotically symmetric metric, defined in a neighborhood of infinity. The geometry of quaternionic contact structures is studied, while octonionic contact structures remain mysterious objects.


Twisteurs des orbites coadjointes et métriques hyper-pseudokählériennes [dvi, ps, pdf]. Bull. Soc. Math. France 126(1), 79-105 (1998).


(avec Paul Gauduchon) La métrique hyperkählérienne des orbites coadjointes de type symétrique d'un groupe de Lie complexe semi-simple [dvi, ps, pdf]. C.R. Acad. Sci. Paris 323, 1259-1264 (1996).


Métriques d'Einstein à cusps et équations de Seiberg-Witten [dvi, ps, pdf]. J. reine angew. Math. 490, 129-154 (1997).


Les équations de Seiberg-Witten sur une surface complexe non kählérienne [dvi, ps, pdf]. Comm. Anal. Geom. 6(1), 173-197 (1998).


(avec Paul Gauduchon) HyperKähler metrics on cotangent bundles of hermitian symmetric spaces [dvi, ps, pdf]. In: Geometry and Physics, J. Andersen, J. Dupont, H. Petersen, A. Swann eds. (Lect. Notes Pure Appl. Math. 184, 287-298), Marcel Dekker, 1997.


(avec Oscar García-Prada) Parabolic vortex equations and instantons of infinite energy [dvi, ps, pdf]. J. Geom. Phys. 21(3), 238-254 (1997).


Fibrés de Higgs et connexions intégrables : le cas logarithmique (diviseur lisse) [dvi, ps, pdf]. Ann. scient. Éc. Norm. Sup. 30(1), 41-96 (1997).


Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes [dvi, ps, pdf]. Math. Ann. 304(1), 253-276 (1996).


Sur les fibrés paraboliques sur une surface complexe [dvi, ps, pdf]. J. London Math. Soc. 53(2), 302-316 (1996).


Prolongement d'un fibré holomorphe hermitien à courbure Lp sur une courbe ouverte [dvi, ps, pdf]. Internat. J. Math. 3(4), 441-453 (1992).


Fibrés paraboliques stables et connexions singulières plates [dvi, ps, pdf]. Bull. Soc. Math. France 119(2), 231-257 (1991).


Fibrés holomorphes et connexions singulières sur une courbe ouverte [dvi, ps, pdf]. Thèse, Ecole Polytechnique, 28 juin 1991.


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Mis à jour le 15 décembre 2010