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# Résultats de recherche

**3322**

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## M. Lesourd - Positive Scalar Curvature on Noncompact Manifolds and the Positive Mass Theorem

The study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in particular the idea to use surfaces of prescribed mean curvature (as opposed to minimal surfaces). Having the classic positive mass theorem of Schoen-Yau in mind, we describe a new positive mass theorem for manifolds that allows for possibly non asymptotically flat ends, points of incompleteness, and regions negative scalar curvature. The proof is ... Voir la vidéole (1h9s)

## D. Stern - Harmonic map methods in spectral geometry

Over the last fifty years, the problem of finding sharp upper bounds for area-normalized Laplacian eigenvalues on closed surfaces has attracted the attention of many geometers, due in part to connections to the study of sphere-valued harmonic maps and minimal immersions. In this talk, I'll describe a series of results which shed new light on this problem by relating it to the variational theory of the Dirichlet energy on sphere-valued maps. Recent applications include new (H^{-1}-)stability results for the maximization of the first and ... Voir la vidéole (1h9s)

## P. Burkhardt - Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C0 initial data. Voir la vidéole (1h14m45s)

## A. Mondino - Time-like Ricci curvature bounds via optimal transport

The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr.After recalling the general setting ... Voir la vidéole (1h2m14s)

## D. Semola - Boundary regularity and stability under lower Ricci bounds

The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem and the Cheeger-Colding theory of Ricci limit spaces. On the other hand “synthetic” theories of lower Ricci bounds have been developed, based on semigroup tools (the Bakry-Émery theory) and on Optimal Transport (the Lott-Sturm-Villani theory). The Cheeger-Colding theory did not consider manifolds with boundary, while in the synthetic framework even understanding what ... Voir la vidéole (1h3m35s)

## C. Li - Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

In this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC manifold of dimension 4 (resp. 5) has vanishing π2 (resp. vanishing π2 and π3), then a finite cover of it is homotopy equivalent to Snor connected sums of Sn-1 x S1. This extends a previous theorem on the non-existence of Riemannian metrics of positive scalar curvature on aspherical manifolds in 4 and 5 dimensions, due to ... Voir la vidéole (1h2m34s)

## Y. Lai - A family of 3d steady gradient Ricci solitons that are flying wings

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed. Voir la vidéole (1h1m56s)

## F. Schulze - Mean curvature flow with generic initial data

Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states ... Voir la vidéole (1h3m55s)