Séminaire de géométrie différentielle

Geometric methods in computational complexity

La géométrie de ces surfaces est conditionnée par une équation de Liouville avec des données au bord de type Dirichlet-Neumann. La géométrie différentielle permet de séparer les variables et de construire des exemples en utilisant les systèmes intégrables. Pour aller plus loin, en combinant des aspects algébriques et analytiques, j’expliquerai comment obtenir des estimées d’énergie pour ces surfaces à bord.

Une structure de produit conforme sur une variété riemannienne $(M,g)$ est une connexion de Weyl (c’est-à-dire une connexion sans torsion qui préserve la classe conforme de la métrique $g$) à holonomie réductible. Nous classifions ces structures dans le cas où $M$ est compacte et $g$ est compatible avec une structure kählerienne. C’est un travail en collaboration avec Mihaela Pilca (Regensburg).

An important family of solutions to the Einstein vacuum equations is given by the Kerr metrics, which describe rotating black holes. In this talk, I will present some important geometric properties of these spacetimes relevant to the study of classical field equations such as the scalar waves, electromagnetism and linearized gravity. As observed by Teukolsky, by exploiting a special algebraic property of the spacetime, it is possible to decouple certain components of the fields from the rest of the system, leading to the so-called Teukolsky equation. Solutions of this equation can then be analyzed to recover information about the full system.

IPVTs and applications

I will discuss limits in low intensity of Poisson-Voronoi tessellations, which we called ideal Poisson-Voronoi tessellations (IPVTs).

In the colloquium part, I will focus on the IPVT of real hyperbolic space of dimension d, where a simple Poissonian description of the cell containing the origin enables an in-depth study of the geometric features of its tiles.

In the research seminar part, I will discuss sufficient conditions for convergence toward IPVTs in a general metric space, and illustrate them for the Cartesian product of two hyperbolic planes endowed with the $L^1$ metric. Then I will discuss an application to proving the smallness of the uniqueness threshold of Poisson/Bernoulli–Voronoi percolation on spaces with a non-amenable product structure.

Based on joint works with Nicolas Curien, Nathanaël Enriquez, Russell Lyons, Meltem Ünel (2303.16831, to appear on The Annals of Probability), on 2412.00822, and on incoming works with Ali Khezeli and with Jan Grebik, Ali Khezeli, Konstantin Recke, and Amanda Wilkens.

The Hawking energy is one of the simplest quasi-local energy definitions in general relativity. Despite its simplicity, the Hawking energy has faced challenges due to ambiguities when applied to general surfaces. In this talk, I will present recent results demonstrating that the Hawking energy exhibits key physical and mathematical properties—non-negativity, rigidity,
and monotonicity—when evaluated on a generalization of  area-constrained Willmore surfaces  (Hawking surfaces). In particular such properties hold for area-constrained Willmore surfaces on manifolds with nonnegative scalar curvature.  These results establish Hawking surfaces as a useful tool for evaluating the Hawking energy and reinforce its potential as a meaningful tool for understanding gravitational phenomena.

The symmetrized Asymptotic Mean Value Laplacian \tilde{\Delta}, is obtained as limit of approximating integral operators \tilde{\Delta}_r, and is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that in the limit as r->0, as the operators eventually admit isolated eigenvalues defined via min-max procedure  on any compact uniformly locally doubling metric measure space. Then we prove L^2 and spectral convergence of \tilde{\Delta}_r to the Laplace-Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.