Riemannian manifolds that are determined by their widths

Differential Geometry Seminar

Organizers：

Jialong Deng, Akito Futaki

Speaker：

Lucas Ambrozio (IMPA)

Time：

Tues., 21:00-22:00, Nov. 26, 2024

Title：

Riemannian manifolds that are determined by their widths

Abstract：

A fascinating subject in Geometric Analysis is whether or not a compact Riemannian manifold can be totally reconstructed from the sole knowledge of its Laplace spectrum. Recent developments in the min-max theory of the area functional on the space of codimension one flat cycles made it possible to ask a similar question regarding the area spectrum. We will present some recent work done in collaboration with A. Neves (UChicago) and F. Marques (Princeton) in that direction. The focus will be on the complete characterization of the class of Riemannian metrics on the three dimensional sphere whose four spherical widths are equal. Time permitting, we will also discuss how these ideas can be further developed, yielding the complete characterization of the real projective plane by its length spectrum.