北京雁栖湖应用数学研究院

The sharp lower bound of the first Neumann eigenvalue on convex domain or compact manifolds with non-negative Ricci curvature had been studied for a long time by Payne-Weinberger (1960’s), Li-Yau and Zhong-Yang (1980’s), later developed by Kroger (1990’s) and Klartag (2010’s). And the corresponding rigidity is obtained by Hang-Wang (2000’s), and also by Klartag (2010’s). We will firstly give a survey on the former results and their methods. Then we will present our recent result about the quantitative stability part of this problem, which links the difference between the first Neumann eigenvalue and its sharp lower bound with the width of convex planar domain. This is a joint work with Haibin Wang, and the talk will be in blackboard.

In this talk, I will talk about the existence of a strictly convex even solution to the Lp prescribed curvature problem. The key idea is to combine a new gradient estimate with Chow-Wang’s geometric lemma to obtain the C^0 and C^2 estimate simultaneously. This is a joint work with M. N. Ivaki.

In this talk, I will introduce some basic concepts about the moduli of Higgs bundles over a curve. Then we consider the surface case and study the moduli spaces via the corresponding Hitchin maps. We try to work out Picard groups of the corresponding spectral surfaces which can be identified as generic fibers of the Hitchin maps. When generic fibers are nonempty, we can say more about the moduli spaces, for example, when the moduli is nonempty. This is a joint work with Bin Wang.

In this talk, we will present the estimation of eigenvalues of the Laplacian operator under Dirichlet and Robin boundary conditions. We will begin by discussing universal inequalities associated with the Laplacian under Dirichlet boundary conditions and their connection to Li-Yau inequalities. We will also talk about the Bossel-Daners inequality of the Laplacian with Robin boundary on Riemannian manifolds. We will explain the inequality in detail and discuss their application. This talk is based on the joint work with Professor Qingming Cheng (Fukuoka University) and Professor Haizhong Li (Tsinghua University).

In this talk, we introduce a new positivity notion for curvature of Riemannian manifolds and obtain characterizations for spherical space forms and the complex projective space. This is a joint work with Prof. Xiaokui Yang.

In this presentation, we will discuss Gromov’s conjecture on the volume bound of Riemannian manifolds with nonnegative Ricci curvature and positive scalar curvature and its variant. As natural analogies, we care about the volume bound and volume growth of Ricci-nonnegative manifolds with positive bi-Ricci curvature and get the optimal bound. This is a joint work with Prof. Jintian Zhu from Westlake University.