Séminaire de géométrie

Certaines EDP géométriques comme l'équation hermitienne de Yang-Mills admettent une interprétation en terme d'application moment, ce qui permet (entre autres) de les étudier d'un point de vue perturbatif. L'idée est la suivante : on part d'une solution connue de cette équation et on en perturbe les paramètres, sous quelles conditions existe-t-il toujours une solution ? Lorsqu'elle existe, comment évolue-t-elle en fonction des paramètres modifiés ? Cet exposé présentera les résultats d'existence et de continuité obtenus ainsi qu'une idée des méthodes utilisées.

Based on joint work with Davide Macera and Joe Thomas. The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil-Petersson probability measure. We show that there is a c>0 such that a random surface of genus g has spectral gap at least 1/4-O(g^-c) with high probability. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil-Petersson random hyperbolic surfaces.

Let $(M,g,X)$ be a complete gradient Kähler–Ricci expander with quadratic curvature decay (including all derivatives). Its geometry at infinity is modeled by a unique asymptotic cone, which takes the form of a Kähler cone $(C0,g0)$. In this talk, we will show that if there exists a solution to the Kähler–Ricci flow on $M$ that desingularizes this cone, then it necessarily coincides with the self-similar solution determined by the soliton metric $g$. Furthermore, if one perturbs the soliton metric in a suitable manner, the resulting initial data generates an immortal solution to the Kähler–Ricci flow which, after appropriate rescaling, converges to an asymptotically conical gradient Kähler–Ricci expander.