Titres et résumés

Letizia Branca

Titre: Bach-pinched metrics on closed four-dimensional manifolds

Résumé:
Exploiting the deformation method introduced by Aubin in his seminal work to con-
struct constant negative scalar curvature metrics, we show the existence, on every closed manifold
of dimension four, of a metric whose Bach tensor is pinched by the scalar curvature. This talk is
based on joint work with G. Catino and D. Dameno.

Bernard Hanke

Titre: Rigidity results for initial data sets satisfying the dominant energy condition

Résumé:

Initial data sets and the dominant energy condition are motivated from general relativity. They can be considered as a generalization of Riemannian manifolds and the condition of positive scalar curvature, respectively.

I will show that a compact initial data set that satisfies the dominant energy condition and a positivity condition for the mean curvature of its boundary can be isometrically embedded in Minkowski space. The image of this embedding can be described explicitly.

The result is sharp. Its proof relies on solving a boundary value problem for the Dirac operator.

This is joint work with Christian Bär, Simon Brendle, and Aaron Chow.

Simon Jubert

Titre: The Yau–Tian–Donaldson correspondence for projective bundles over curves

Résumé:

Following remarkable advances in algebraic and analytic geometry over the past decades, recent deep work by Boucksom–Jonsson has established the Yau–Tian–Donaldson conjecture for projective manifolds. This result proves the equivalence between the existence of a canonical Kähler metric and an algebro-geometric notion known as K-stability. Although this bridge reduces the existence problem for a geometric PDE to an algebraic condition, K-stability remains extremely difficult to test on a generic variety. Over the years, several strategies have been developed to make stability more tractable. In this talk, I will present a resolution of the conjecture for the projectivization of a holomorphic vector bundle over a curve. On the analytic side, we will see that one has a precise description of the canonical metrics in terms of the topological invariants of the fibration and a suitable Kähler metric on its fiber. On the algro-geometric side, I will explain how the symmetries of the manifold can be exploited to fully translate the K-stability as a convex geometric condition on a suitable moment polytope. This is joint work (in progress) with Chenxi Yin (UQAM).

Alan Pinoy

Titre : A new positive mass theorem for asymptotically hyperbolic 3d manifolds via Green function

Résumé :
(joint work with Klaus Kröncke and Francesca Oronzio)

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 work, we prove an analogous positive mass theorem for this quantity in the 3 dimensional case. Precisely, we show that the volume-renormalised mass of an asymptotically hyperbolic 3d manifold 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.

Annachiara Piubello

Titre: A geometric choice of asymptotically Euclidean coordinates via STCMC-foliations.

Résumé:
Asymptotically Euclidean initial data sets (IDS) in General Relativity model instants in time for isolated systems. In this talk, we show that an IDS is asymptotically Euclidean if it admits a cover by closed hypersurfaces of constant spacetime mean curvature (STCMC), provided these hypersurfaces satisfy certain geometric estimates, some weak foliation properties, and each surface exhibits generalized stability. Building on the work of Cederbaum and Sakovich (2021), which established that every asymptotically Euclidean IDS has a unique STCMC foliation, we conclude that the existence of such a foliation characterizes asymptotically Euclidean IDS. Furthermore, we explore the connections to the center of mass and show why these coordinates seem well-adapted to describe this concept. This is joint work with O. Vičánek Martínez.

Thomas Richard

Titre: Smoothing of cyclic 3-orbifolds with positive scalar curvature

Résumé:
A Riemannian orbifold (O,g) endows the underlying topological space |O| with a distance d which does not come from a smooth Riemannian metric on |O|. In dimension 3, the singularities of this distance on a trivalent graph ∑, the (O^3) is said to be cyclic if ∑ is a collection of disjoint curves. In this talk we will show if a cyclic Riemannian (O^3,g) has postive scalar curvature then the metric space (|O|^3,d) can be smoothed with positive scalar curvature. This smoothing is built in a very hands-on manner which relies on a normal form for the orbifold metric near singularities together with a metric modification procedure pioneered by Aubin.

Rudolf Zeidler

Titre: Spin geometry and scalar curvature rigidity

Résumé:

The goal of this mini‑course is to introduce spinorial methods for proving rigidity phenomena under scalar curvature bounds. We begin with preliminaries such as Clifford algebras and spin structures on vector bundles. We then develop the Dirac operator and review its key properties needed to study scalar curvature problems. Building on this, we explain classical obstructions to the existence of positive scalar curvature metrics on closed spin manifolds and Llarull's extremality and rigidity theorem.

In the second lecture (modulo precise timing), we will give a quick introduction to boundary value problems for Dirac operators and their application to scalar-mean rigidity theorems on manifolds with boundary. In particular, we will study strictly convex domains in Euclidean space and Gromov’s hyperspherical radius rigidity (based on joint work with Cecchini and Hirsch).

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