## Flot par l'inverse de la courbure moyenne dans l'espace hyperbolique complexe

Type:
Site:
Date:
06/03/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
PIPOLI Giuseppe
Localisation:
Université Grenoble Alpes
Localisation:
France
Résumé:

Nous considérons l'évolution par l'inverse de la courbure moyenne d'une surface étoilée, fermée et à courbure moyenne positive dans l'espace hyperbolique complexe. Nous montrons que le flot est défini pour tout temps positif et que la surface reste étoilée et à courbure moyenne positive. De plus, la métrique induite, après un changement d'échelle, converge vers un multiple conforme de la métrique sous-riemannienne standard sur la sphère de dimension impaire. Nous allons montrer l'existence d'exemples de données initiales telles que cette limite sous-riemannienne n'a pas la courbure de Webster constante.

## Polyhedral surfaces in Cauchy-compact $3$-dimensional flat spacetimes with BTZ-like singularities with help from Teichmüller

Type:
Site:
Date:
20/02/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
BRUNSWIC Léo
Localisation:
Université d'Avignon
Localisation:
France
Résumé:

In the 1990's, T'Hooft suggested to study 3-dimensional singular flat spacetimes with polyhedral Cauchy-surfaces as toy model to understand quantum gravity. This motivates the study of singular spacetimes however the type of a singularity in a Lorentzian manifold depends on both the type of the axis and the causality around it which strongly contrast with the riemannian context. BTZ-like singularities are limit cases of "massive particles" which are close Lorentzian equivalent to conical singularities.

We present some classification results on Cauchy-compact spacetimes with BTZ and present ramifications of the convex hull method used by Penner to construct a cellulation of his decorated Teichmüller space.

## Constant curvature surfaces in $(2+1)$-Minkowski space

Type:
Site:
Date:
06/02/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
SEPPI Andrea
Localisation:
Université de Pavie
Localisation:
Italie
Résumé:

We will discuss the problem of existence and uniqueness of surfaces of negative constant (or prescribed) Gaussian curvature $K$ in $(2+1)$-dimensional Minkowski space. The simplest example, for $K=-1$, is the well-known embedding of hyperbolic plane as the one-sheeted hyperboloid; however, as a striking difference with the sphere in Euclidean space, in Minkowski space there are many non-equivalent isometric embeddings of the hyperbolic plane.

This problem is related to solutions of the Monge-Ampère equation $\det D^2 u(z)=(1/|K|)(1-|z|^2)^{-2}$ on the unit disc. We will prove the existence of surfaces with the condition $u=f$ on the boundary of the disc, for $f$ a bounded lower semicontinuous function. If the curvature $K=K(z)$ depends smoothly on the point $z$, this gives a solution to the so-called Minkowski problem.

On the other hand, we will prove that, for $K$ constant, the principal curvatures of a $K$-surface are bounded from below by a positive constant if and only if the corresponding function $f$ is in the Zygmund class. Time permitting, we will discuss some generalizations to constant affine curvature.

## On the evolution by fractional mean curvature

Type:
Site:
Date:
30/01/2017 - 13:30 - 30/07/2017 - 14:30
Salle:
2015
Orateur:
SAEZ Mariel
Localisation:
Université pontificale catholique
Localisation:
Chili
Résumé:

In this work we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric objects that in turn yield the preservation of certain quantities, such as the positivity of the fractional mean curvature.

## On complete maximal hypersurfaces in Robertson-Walker spacetimes with flat fiber

Type:
Site:
Date:
23/01/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
SANCHEZ PELEGRIN José Antonio
Localisation:
Localisation:
Espagne
Résumé:

In this talk, we will deal with complete maximal hypersurfaces in spatially open $(n+1)$-dimensional Robertson-Walker spacetimes with flat fiber. Indeed, under natural geometric and physical assumptions we will obtain a new Calabi-Bernstein type result for these hypersurfaces as well as some nonexistence ones. To conclude, we will also apply these results to some relevant spacetimes, such as the steady state spacetime, the Einstein-de Sitter spacetime and certain radiation models.

## Trapped submanifolds in de Sitter space

Type:
Site:
Date:
16/01/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
ALIAS Luis
Localisation:
Université de Murcie
Localisation:
Espagne
Résumé:

The concept of trapped surfaces was originally formulated by Penrose for the case of 2-dimensional spacelike surfaces in $4$-dimensional spacetimes in terms of the signs or the vanishing of the so-called null expansions. This is obviously related to the causal orientation of the mean curvature vector of the surface, which provides a better and powerful characterization of the trapped surfaces and allows the generalization of this concept to codimension two spacelike submanifolds of arbitrary dimension $n$. In this sense, an $n$-dimensional spacelike submanifold $\Sigma$ of an $(n + 2)$- dimensional spacetime is said to be future trapped if its mean curvature vector field $H$ is timelike and future-pointing everywhere on $\Sigma$, and similarly for past trapped. If $H$ is lightlike (or null) and future-pointing everywhere on $\Sigma$ then the submanifold is said to be marginally future trapped, and similarly for marginally past trapped. Finally, if $H$ is causal and future-pointing everywhere, the submanifold is said to be weakly future trapped, and similarly for weakly past trapped. The extreme case $H = 0$ on $\Sigma$ corresponds to a minimal submanifold. In this lecture we consider codimension two compact marginally trapped submanifolds in the light cone of de Sitter space. In particular, we show that they are conformally diffeomorphic to the round sphere and, as an application of the solution of the Yamabe problem on the round sphere, we derive a classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space.

This is part of our work in progress with Verónica L. Cánovas (from Murcia) and Marco Rigoli (from Milano).

## Une caractérisation spectrale des boules géodésiques dans les espaces symétriques de rang un

Type:
Site:
Date:
09/01/2017 - 13:30 - 14:30
Salle:
2015
Orateur:
CASTILLON Philippe
Localisation:
Université Montpellier 2
Localisation:
France
Résumé:

Dans les géométries de courbure constante, les boules géodésiques sont les domaines optimaux pour de nombreux problèmes d'optimisation de formes, notamment de nature spectrale. On peut s'attendre à des caractérisations similaires dans les espaces symétriques de rang un, dans la mesure où les boules y sont les domaines les plus symétriques. Cependant très peu de résultats de ce type y sont connus.

Dans cet exposé nous montrerons que les boules géodésiques sont les seuls maximiseurs de la première valeur propre de Steklov parmi les domaines de volume fixé, généralisant aux espaces symétriques de rang un non compacts une inégalité obtenue par F. Brock dans l'espace euclidien.

Il s'agit d'un travail en commun avec Berardo Ruffini.

## Contractibilité de courbes sur le bord des $3$-variétés

Type:
Site:
Date:
12/12/2016 - 13:30 - 14:30
Salle:
2015
Orateur:
COLIN de VERDIERE Eric
Localisation:
Université de Marne-la-vallée
Localisation:
France
Résumé:

Etant donnée une courbe $c$ dans une $3$-variété triangulée $M$, comment déterminer si $c$ est contractile ?

Dans la suite, nous supposons toujours que $c$ est sur le bord de $M$. Le cas où $c$ est sans auto-intersections a été étudié par Hass, Lagarias et Pippenger (1999) en utilisant la notion de surfaces normales, en lien avec le problème du noeud ; ils montrent que le problème est dans NP, ce qui donne un algorithme exponentiel. Je décrirai un algorithme avec la même complexité qui résout le problème dans le cas plus général où $c$ peut avoir des auto-intersections. La méthode repose de façon clé sur la démonstration du Loop Theorem.

Cet exposé, résultat d'un travail en commun avec Salman Parsa, ne nécessite aucune connaissance préalable en algorithmique et complexité.

## Noncompact self-expanding solitons of inverse mean curvature flow

Type:
Site:
Date:
05/12/2016 - 13:30 - 14:30
Salle:
2015
Orateur:
LEE Hojoo
Localisation:
KIAS
Localisation:
Corée du sud
Résumé:

While the round spheres are exceptionally rigid among compact self-expanding solitons, even in the class of rotational hypersurfaces, there are interesting examples of complete noncompact self-expanders. Indeed, G. Huisken and T. Ilmanen constructed a complete, rotationally symmetric, self-expander with one asymptotically cylindrical end. First, we use the shooting method to construct new self-expanders, so called infinite bottles, having two cylindrical ends (joint with G. Drugan and G. Wheeler, 2015). Second, motivated by the role of Jacobi fields for constant mean curvature surfaces, we investigate the linearized operator of the soliton equation to establish the rigidity of rotational self-expanders (joint work with G. Drugan and F. Fong, 2016).

## The Ricci Flow on manifolds with almost non-negative curvature operator

Type:
Site:
Date:
28/11/2016 - 13:30 - 14:30
Salle:
2015
Orateur:
CABEZAS-RIVAS Esther
Localisation:
Université de Francfort
Localisation:
Allemagne
Résumé:

We show that $n$-manifolds with a lower volume bound $v$ and upper diameter bound $D$ whose curvature operator is bounded below by $-\varepsilon(n,v,D)$ also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound.