Spherical Plateau Problem and Applications in Geometric Analysis

Explore a lecture on the Spherical Plateau problem and its applications, delivered by Antoine Song from Caltech at the Institut des Hautes Etudes Scientifiques (IHES). Delve into an area minimization problem in certain quotients of the Hilbert sphere by countable groups, examining its origins in Besson-Courtois-Gallot's work on entropy inequality. Discover stability results derived from this minimization problem, including an analysis of closed surfaces of genus at least 2 with Riemannian metrics. Learn about the relationship between volume entropy, first eigenvalue, and hyperbolic metrics, and understand the stability of the hyperbolic plane in Benjamini-Schramm topology. Follow the lecture's outline, covering topics such as Gamma-compactness theorem, geometrization, hyperbolic manifolds, volumetropy inequality, and convergence over the course of one hour.

Outline
Problem
Gamma
Compactness Theorem
Motivation
Questions
Geometricization
Hyperbolic manifolds
Open questions
Volumetropy
Inequality
Statement
Topology
Convergence