Titres et résumés

Unknottedness of free boundary minimal surfaces and self-shrinkers (Giada Franz)

Lawson in 1970 proved that minimal surfaces in the three-dimensional sphere are unknotted. In this talk, we discuss unknottedness of free boundary minimal surfaces in the three-dimensional unit ball and of self-shrinkers in the three-dimensional Euclidean space.This is based on joint work with Sabine Chu.

Weak immersions with second fundamental form in a critical Sobolev space (Dorian Martino)

Over the past decade, the generalization of the Willmore energy to even dimensional submanifolds of the Euclidean space has been subject to numerous works, due to their relation to renormalized volume of minimal submanifolds in Poincaré-Einstein manifolds and their applications to AdS/CFT correspondence.

I will present an analytical framework, recently developed in a collaboration with Tristan Rivière, in order to address variational problems concerning these generalized Willmore energies.

Optimization of Eigenvalues of GJMS Operators in a Conformal Class (Romain Petrides)

On a Riemannian manifold of dimension $n \geq 3$, for every positive integer $k$, there exists a conformal covariant differential operator of even order $2k \leq n$, whose leading term is the $k$-th power of the Laplacian (GJMS operator). For $k = 1$, this is the famous conformal Laplacian that appears in the Yamabe problem. We consider the more general problem of minimizing (resp. maximizing) the positive eigenvalues (resp. negative eigenvalues) of these operators among all metrics with fixed volume in a given conformal class, in the case $2k < n$. In particular, we calculate optimal bounds and provide examples where they are attained, as well as others where they are not, depending on the choice of the manifold, the conformal class, $k$, and the index of the eigenvalue. At the interface of spectral geometry and conformal geometry, this is a joint work with E. Humbert and B. Premoselli.

A new positive mass theorem for 3 dimensional asymptotically hyperbolic manifolds via potential theory (Alan Pinoy)

The (Euclidean) positive mass theorem plays a central role in geometric analysis and mathematical general relativity. It characterises the Euclidean space among asymptotically Euclidean manifolds of non-negative scalar curvature as the unique minimiser of the ADM mass, which is non-negative. Recently, Dahl-Kröncke-McCormick proposed and studied a new geometric quantity for asymptotically hyperbolic manifolds, called the volume-renormalised mass, which is a linear combination of the usual notions of the ADM mass and the renormalised volume. In this talk, after a review of some properties of this new mass quantity, we prove an analogous positive mass theorem in the 3 dimensional case. Precisely, we show that the volume-renormalised mass of an asymptotically hyperbolic 3 dimensional manifold (with a mild topological condition) and with scalar curvature greater than 6 is non-negative, and vanishes only for the hyperbolic space. To do so, we prove a new monotonicity formula that holds along the level sets of the minimal Green function of such a manifold. This is a joint work with Klaus Kröncke (KTH Stockholm) and Francesca Oronzio (SNM Napoli).

A conformal approach to the existence of constant scalar curvature Kähler metrics (Carlo Scarpa)

One of the fundamental problems in geometric analysis is the existence of Riemannian metrics with constant scalar curvature. In the 1980s, the celebrated solution of the Yamabe problem by Trudinger, Aubin, and Schoen established that on a closed manifold one can always find a constant scalar curvature metric, in each conformal class of Riemannian metrics. In the context of Kähler manifolds however, the problem is still largely open: conformal transformations typically do not preserve the Kähler condition. In this talk, I will present some progress in an ongoing project with Abdellah Lahdili (Université du Québec à Montréal) and Eveline Legendre (Université Lyon 1), in which we propose an approach to the existence of constant scalar curvature Kähler metrics using tools and techniques that first appeared in the solution of the Yamabe problem in CR geometry. In particular, I will explain how one can use the CR Yamabe invariant to detect the existence of constant scalar curvature Kähler metrics.

An overview of Kato limit spaces (David Tewodrose)

I will report on a series of joint papers with Gilles Carron (Nantes Université) and Ilaria Mondello (Université Paris-Est Créteil), in which we investigate the geometric and analytic properties of Kato limit spaces. These are defined as Gromov-Hausdorff (GH) limits of complete Riemannian manifolds satisfying uniform Kato bounds on the Ricci curvature. After reviewing these Kato bounds, I will provide a simple 2D example of a branching Kato limit space that cannot be GH-approximated by manifolds with a uniform Ricci curvature lower bound. I will also explain why this example does not lie in the GH closure of the set of manifolds satisfying a small Ricci curvature Lp bound in the sense of Petersen-Wei.

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