Titles and abstracts

Mini-course: Low regularity problems bridging Analysis & Geometry (Esther Cabezas-Rivas)

We will provide examples of past, present and future work on geometric flows (like inverse mean curvature flow or Ricci flow) and manifold-constrained minimization problems with applications to image processing; all of them under mild regularity assumptions. We will focus on how to explore existence and uniqueness issues, as well as if the solutions get smoother or keep the regularity of the source or initial data. The idea is to devote the first half of the mini-course to background material for non-experts, and give details about current research work in the last sessions.

Best constants in Hardy-Poincaré-Sobolev inequalities for some weightedhyperbolicspaces (Baptiste Devyver)

In this talk, we will consider optimal Sobolev inequalities with their best constant on some weighted Riemannian manifolds with positive bottom of the spectrum of the Laplacian. The question we will investigate is whether it is possible to control the energy, shifted by the first eigenvalue, by an Lp norm, keeping a ''Euclidean'' optimal constant. The work of Benguria-Frank-Loss and Mancini-Sandeep (2007) shows that in the case of usual hyperbolic space of dimension 3, such an optimal Sobolev inequality indeed holds. On the other hand, such a phenomenon do not occur for dimension at least 4. Results on this problem will be presented in the context of some ''weighted'' hyperbolic spaces which appear naturally in the study of Caffarelli-Kohn-Nirenberg inequalities. It is a joint work with Pierre-Damien Thizy and Louis Dupaigne from Université Lyon 1.

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.

Z_2-harmonic forms and branched deformations of co-associatives in G2-manifolds (Thibault Langlais)

G2-manifolds form an exceptional class of Ricci-flat 7-manifolds, whose tangent bundle is endowed with a parallel cross-product modelled on the cross-product of the imaginary octonions. An interesting feature of G2-geometry is the existence of distinguished classes of calibrated submanifolds, namely associatives (codimension 4) and co-associatives (codimension 3), which are analogous to complex and Special Lagrangian submanifolds of Calabi—Yau manifolds. Due to a classical result of McLean, the infinitesimal deformations of these calibrated submanifolds are parametrised by harmonic spinors (in the associative case) and by harmonic self-dual 2-forms (in the co-associative case). The goal of this talk is to present joint work-in-progress with Jacek Rzemieniecki to extend this result to branched deformations of co-associatives, in relation with the recent work of Siqi He on Special Lagrangians. The infinitesimal branched deformations are modelled on the so-called Z_2-harmonic forms, which were originally introduced in order to compactify the moduli spaces of solutions to the Vafa—Witten equations.

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.

Conformal plane waves and the Lorentzian Lichnerowicz conjecture (Lilia Mehidi)

It is a classical fact that the isometry group of a compact Riemannian manifold is a compact Lie group. In contrast, the conformal group need not be compact. However, it is compact in all cases except one: the standard sphere, whose conformal group is non-compact. This fact was conjectured by Lichnerowicz and proved by Ferrand and Obata in the 1970s. Equivalently, the sphere is the only compact Riemannian manifold whose conformal group does not preserve any Riemannian metric.
The Lorentzian version of the Lichnerowicz conjecture (LLC) asserts that a compact Lorentzian manifold whose conformal group is not isometric for any metric in the conformal class must be conformally flat. This conjecture remains open, although significant progress has been made in certain cases.
We propose to address this question in a locally conformally homogeneous setting. Thanks to a recent result of Alekseevsky and Galaev, it turns out that certain deformations of Minkowski space, called plane waves (which are generically non-conformally flat), play an important role in the study of the LLC.

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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