ma bobine

Philippe Castillon

Research topics

My research concerns Riemannian geometry and global analysis, in general on non-compact manifolds. In particular I'm working on the following topics: optimal transport on Riemannian manifolds, spectral theory of geometric operators, isoperimetric and functional inequalities, asymptotic geometry in negative curvature.

Geometry and optimal transport of measures

In a work in progress, I'm considering the problem of prescription of the Gauss curvature for convex bodies. In the Euclidean setting, the use of optimal transportation gave a new approach for this problem initially considered by Alexandrov. For convex bodies in the hyperbolic space or in the sphere, this question of finding a convex body with a given Gauss curvature is still natural. With J. Bertrand we are considering this non-Euclidean setting, using the optimal transport approach.

Isoperimetric inequalities (and their analytic counterparts, the Sobolev inequalities) are commonly used in global analysis. These inequalities are interesting for their own, a widely studied problem being to get sharp constant.  In the Euclidean space, the use of optimal transport map gave a new approach of these problems. I'm working on transposing this approach to non-Euclidean settings, in particular for submanifolds.

Spectral theory of geometric operators

The geodesic balls in constant curvature spaces are known to be optimal domains for several geometric quantities, some of them involving eigenvalues of geometric operators. Although it is natural to expect such characterizations in spaces where geodesic spheres have a lot of symmetries (like in rank one symmetric spaces), very few is known in this case. With B. Ruffini we are considering, in this setting, the problem of maximizing the first Steklov eigenvalue among domains of fixed volume.

Minimal and constant mean curvature hypersurfaces are critical points of the volume functional. I'm interested in the stability operators associates to these variational problems which are Schrödinger operators whose potential depends on the curvatures of the submanifold and the ambient manifold. A natural problem in this setting is to get topological and/or geometrical properties from the non-negativity of such operators.

Harmonic and asymptotically harmonic manifolds

My work on constant mean curvature hypersurfaces led me to work on harmonic manifolds (which are those Riemannian manifolds whose spheres have constant mean curvature) for which many natural questions remain open, such as the isoperimetric problem.

Asymptotically harmonic manifolds are those Riemannian manifolds whose horospheres have constant mean curvature. The mean curvature of the horospheres is an asymptotic invariant which can be related to other classical invariants (volume entropy, essential spectrum, harmonic and visual measures).


Publications


Published papers

  1. J. Bertrand, P. Castillon. Prescribing the Gauss curvature of convex bodies in hyperbolic space, arXiv:1903.06502.

  2. P. Castillon, B. Ruffini. A spectral characterization of geodesic balls in non-compact rank one symmetric spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (to appear), arXiv:1611.06081.

  3. P. Bérard, P. Castillon. Inverse spectral positivity for surfaces, Rev. Mat. Iberoam., 30 (2014), 1237-1264.

  4. P. Castillon, A. Sambusetti. On asymptotically harmonic manifolds of negative curvature, Math. Z., 277 (2014) 1049-1072.

  5. P. Bérard, P. Castillon. Spectral positivity and Riemannian covering, Bull. London Math. Soc, 45 (2013), 1041-1048.

  6. P. Bérard, P. Castillon, M. Cvalcante. Eigenvalue estimates for hypersurfaces in Hn×R and applications, Pacific J. Math., 253 (2011), 19-35.

  7. P. Castillon. Submanifolds, isoperimetry and optimal transportation, J. Funct. Anal., 259 (2010), 79-103.

  8. P. Castillon. An inverse spectral problem on surfaces, Comment. Math. Helv., 81 (2006), 271--286.

  9. P. Castillon. Problèmes de petites valeurs propres sur les surfaces de courbure moyenne constante, Proc. Amer. Math. Soc., 130 (2001), 1153-1163.

  10. P. Castillon. Spectral properties of constant mean curvature submanifolds in hyperbolic space, Ann. Global Anal. Geom., 17 (1999), 563-580.

  11. P. Castillon. Sur les surfaces de révolution à courbure moyenne constante dans l'espace hyperbolique, Ann. Fac. Sci. Toulouse, 7 (1998), 379-400.

  12. P. Castillon. Sur l'opérateur de stabilité des sous-variétés à courbure moyenne constante dans l'espace hyperbolique, Manuscripta Math., 95 (1997), 385-400.

Notes, proceedings of seminars

  1. P. Bérard, P. Castillon, Remarks on J. Espinar's "Finite index operators on surfaces", arXiv:1204.1604.

  2. P. Castillon. Un problème spectral inverse sur les surfaces, Sémin. Théor. Spectr. Géom., 20 (2001-02), 139-142.

  3. P. Castillon. Spectral properties and conformal type of surfaces, An. Acad. Bras. Ciênc. 74 (2002), 585-588.

  4. P. Castillon. Métriques à entropie topologique positive sur S2, Sémin. Théor. Spectr. Géom., 10 (1991-92), 97-107.

Thesis, memoir

  1. Analyse globale et sous-variétés, mémoire de synthèse, habilitation à diriger les recherches, 2010.

  2. Sur les sous-variétés à courbure moyenne constante dans l'espace hyperbolique, PhD, 1997.


Conferences (the most recent ones)



-- Last modified, September 2019 --