报告题目：Eigenvalue estimates and a conjecture of Yau

报告摘要：I will describe various upper and lower bounds on the spectrum of the Laplace-Beltrami (and associated Schrödinger) operator on Riemannian manifolds. The upper bounds lead to some important results in spectral geometry establishing a link between the so-called conformal spectrum (and extremal metrics) and branched minimal immersions into Euclidean spheres. I will as well describe some estimates on spectral projections, which amount to understand some concentration properties of eigenfunctions. Derivation of those various bounds are aimed at understanding the geometry and topology (and dynamical properties) of manifolds in terms of spectral data.

I will then describe a conjecture by Yau on the first eigenvalue on minimal submanifolds of the sphere, which is known only for some examples and still largely open. I will then present some recent results where we improve quantitatively the best known lower bound (in the general case) of Choi and Wang of the mid 80’s. I will address some open problems and possible directions to generalize the results.

报告人简介：Yannick Sire，法国人，现任美国约翰霍普金斯大学数学系终身教授、副主任（分管研究生教学），在偏微分方程、调和分析、动力系统等领域做出了一系列杰出的工作。2018年入选西蒙斯基金会数学会士，2021年入选美国数学会会士。

报告时间：2025年7月28日16：00-17:00

报告地点：同析4号楼412会议室