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Complex Analysis and Geometry Seminar

A new functional for extremal metrics in high dimensions

Yannick Sire, Johns Hopkins University

Location:  Room 705
Date & time: Friday, 24 January 2020 at 10:30AM - 11:30AM

Abstract:  Extremal metrics for the spectrum of the Laplace-Beltrami operator on  Riemannian surfaces are important objects in geometry and topology to understand a surface out of its (conformal) spectrum. I will report on a recent work with Hang Xu (UCI) about a new functional for the conformal spectrum of the conformal laplacian on a closed manifold of dimension at least. For this functional, we proved a Korevaar type result. Before doing so, I will explain what is known in the case of surfaces about existence and regularity of the extremal metric, as well as some isoperimetric inequalities for special manifolds. Even if this topic has seen a lot of progress in the last years, there are still many open questions that I will emphasize. In particular, there are many important connections with the theory of minimal immersions into spheres and algebraic geometry.


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