报告信息

陈志杰 (清华大学)

Mean Field Equations and Spectrum of the Associated Complex Hill Operator

I will talk about mean field equations with singularities on flat tori and the associated complex Hill operator with the Treibich-Verdier potential. We will introduce a necessary and sufficient condition, which comes from the study of the mean field equation, to guarantee that the spectrum is of finite band type.

黄红（北京师范大学）

Compact Manifolds with Positive Isotropic Curvature

In this talk I will first recall some results known before 2017 about manifolds with positive isotropic curvature, then I will briefly introduce Brendle's recent important work on Ricci flow on compact manifolds of dimension $n>11$ with positive isotropic curvature, finally I will describe some attempts of the speaker to extend Brendle's work.

郑涛（北京理工大学）

The Dirichlet Problem of Fully Nonlinear Equations on Hermitian Manifolds

We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradients of the solutions. In some special cases, we also obtain the gradient estimates, and hence we can solve the corresponding Dirichlet problem with admissible subsolutions, which is mainly motivated by our alternative proof of the upper bound of the gradients of the solutions to the equations related to the $(m-1)$-plurisubharmonic functions solved by Tosatti \& Weinkove and to the Gauduchon conjecture solved by Sz\'ekelyhidi, Tosatti \& Weinkove on the compact Hermitian manifolds without boundary.This is a joint work with Dr. Ke Feng and Professor Huabin Ge.

刘双（中国人民大学）

Li-Yau Inequality and its Applications on Graphs

This talk is concerning Li-Yau inequality and its applications on graphs, including bounded and unbounded Laplacians. Li-Yau inequality is a powerful tool for studying positive solutions to the heat equation. By studying the heat semigroup, we derive Li-Yau inequality with the assumption of curvature-dimension inequality CDE'(n,K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, such as Harnack inequality, heat kernel bounds. This talk is based on a joint work with Chao Gong, Yong Lin and Shing-Tung Yau, and a joint work with Pual Horn, Yong Lin and Shing-Tung Yau.

魏国栋（北京大学）

On the Fill-in of Nonnegative Scalar Curvature Metrics

A triple of (generalized) Bartnik data $(\Sigma^{n-1},\gamma,H) $ consists of an closed Riemannian manifold $(\Sigma^{n-1},\gamma)$ and a given smooth function $H$ on $\Sigma^{n-1}$. One basic problem in Riemannian geometry is to study“under what kind of conditions does the Bartnik data $(\Sigma^{n-1},\gamma,H) $ admit a fill-in metric $g$ with scalar curvature bounded below by a given constant?” That is, there are a compact Riemannian manifold $(\Omega^n , g )$ with boundary of scalar curvature $R_g\geq \sigma>-\infty$, and an isometry $X: (\Sigma^{n-1}, \gamma)\mapsto (\partial \Omega^n, g|_{\partial \Omega^n})$ so that $H=H_g \circ X$ on $\Sigma$, where $H_g$ is the mean curvature of $\partial \Omega^n$ in $(\Omega^n, g)$ .We prove that given a metric $\gamma$ on $\mathbf{S}^{n-1}$ ($3\leq n\leq 7$), $(\mathbf{S}^{n-1},\gamma,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough. Moreover, we prove that if $\gamma$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $\mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $\int_{\mathbf S^{n-1}}H\,\mathrm d\mu_\gamma$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov. This talk is based on my recent work joint with Prof. Yuguang Shi, Dr. Wenlong Wang and Dr. Jintian Zhu.