# 微分几何与几何分析讨论班(2018秋季学期)

### 地点：蒙民伟楼1105报告厅

 时 间 地 点 报告人 题目与摘要 2018年9月21日10:00–12:00 蒙民伟楼1105室 张希(中国科技大学) Title: Nonlinear PDE in complex geometric analysis Abstract: Hermitian Yang-Mills equation and Complex Monge-Amp\`ere equation are two very important two nonlinear PDE in Complex Geometry. In this talk， I will recall some classical results, and introduce our recent works on these two nonlinear PDE and their applications in complex geometry. These works are joint with Dinew, Guan Pengfei, Jin Xi shen, Li Chao, Li Jiayu, Liu Jiawei,  Zhang Chuanjing,  Zhang Pan and Zhang Xiangwen. 2018年9月28日10:00-12:00 蒙民伟楼1105室 王林峰(南通大学) Title: p-Laplace parabolic equation on manifolds and graphs Abstract:  In the first part of this report, by a regularization process we derive a new gradient estimate for the p-Laplace parabolic equation on a closed manifold with the Ricci curvature bounded from below by a negative number, which includes the gradient estimate established by Ni and Kotschwar on closed manifolds with nonnegative Ricci curvature, and also includes the Davies, Hamilton, and Li-Xu's gradient estimates. In the second part of this report, we establish a general gradient estimate for the p-Laplace parabolic equation on a connected finite graph under a suitable curvature-dimension condition. When the curvature is nonnegative we derive the logarithmic gradient estimate; when the curvature is bounded from below by a negative number we derive the Davies, Hamilton, Bakry-Qian and Li-Xu's estimate, as special cases. Based on the gradient estimates, we derive the Harnack inequalities. 2018年10月12日10:00–12:00 蒙民伟楼1105室 刘世平(中国科技大学) Title: What are discrete spheres? Abstract: The Bonnet-Myers theorem states that an n-dimensional complete Riemannian manifold M with Ricci curvature lower bounded  by a positive number (n-1)K is compact, and its diameter is no greater than $\pi/\sqrt{K}$. Moreover, Cheng’s rigidity theorem tells that the diameter estimate is sharp if and only if M is the n-dimensional round sphere. In this talk, I will discuss discrete analogues of round spheres in graph theory via exploring discrete Bonnet-Myers-Cheng type results. This talk is based on joint works with Cushing, Kamtue, Koolen, Muench, and Peyerimhoff. 2018年10月19日10:00—12:00 蒙民伟楼1105室 盛为民(浙江大学) Title: FLOW BY GAUSS CURVATURE TO THE ALEKSANDROV AND DUAL MINKOWSKI PROBLEMS Abstract: In this talk, I will introduce my work on Gauss curvature flow with Xu-Jia Wang and Qi-Rui Li. In this work we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\R^{n+1}$ with the speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. We prove that if $\alpha\ge n+1$, the flow exists for all time and converges smoothly after normalization to a hypersurface, which is a sphere if $f\equiv 1$.  Our  argument provides a new proof for the classical Aleksandrov problem  ($\alpha = n+1$) and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang recently, for the case $q<0$ ($\alpha>n+1$). If $\alpha< n+1$, corresponding to the case $q > 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence. 2018年10月19日14:00—16:00 蒙民伟楼1105室 李海中(清华大学) Title: Variational problems for Riemannian functionals and the generalized Willmore conjecture Abstract: In this talk, we present our research works about variational problems of Riemannian functionals on an $n$-dimensional compact Riemmannian manifold $(M,g)$, which include the renormalized volume coefficients functional $\int_M v^{2k}(g)dv_g$, and the Weyl curvature functional $\int_M |W(g)|^{n/2}dv_g$. For a hypersurface in a sphere, we study the generalized Willmore functional and generalized Willmore conjecture. By use of an inequality between the Weyl curvature functional and the generalized Willmore functional, we give some discussions about the Generalized Willmore conjecture for 4-dimensional compact  hypersurfaces in a sphere. 2018年10月26日10:00—12:00 蒙民伟楼1105室 傅吉祥(复旦大学) Title: A survey on balanced metrics Abstract: In this talk, I will give a survey on balanced metrics, including the existence of balanced metrics, the solutions to the Strominger system, small deformation of balanced metrics, and the balanced cone of a Kahler manifold. 2018年11月2日10:00—12:00 蒙民伟楼1105室 陈世炳(中国科技大学) Title: On the second boundary value problem for Monge-Ampere equation Abstract: I will talk about the global smoothness of solutions to the  Monge-Ampere equation with the second boundary condition. Besides its important connection to the optimal transport problem, it has many interesting applications in geometric problems such as prescribing Gauss curvature problem and minimal Lagrangian graphs. The talk is based on joint works with Jiakun Liu and Xu-Jia Wang. 2018年11月10日14:00—16:00 蒙民伟楼1105室 朱晓宝(中国人民大学) Title: Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case Abstract: In this talk, we shall present a new result about prescribing Gaussian curvature on a closed Riemann surface with conical singularities in the negative case. This is a joint work with Professor Yunyan Yang. 2018年11月16日10:00—12:00 蒙民伟楼1105室 马力(北京科技大学) Title: Translating mean curvature flow of hypersurfaces Abstract: In this talk, we discuss a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in Rn+1. We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered. If time permits, we discuss some open questions. 2018年11月23日 10:00–12:00 蒙民伟楼1105室 方益(安徽工业大学) Title: Applications of Brown York mass on Besse’s conjecture and related topics Abstract: In Arthur L. Besse’s book “Einstein manifolds”, one of interesting problems therein is about Hilbert-Einstein functional. It was conjectured that the critical points of Hilbert-Einstein functional with unit volume and constant scalar curvature have to be Einstein. This is usually referred to be Besse’s conjecture. It still remains open today. In this talk, we will show a connection between Besse’s conjecture and positive mass theorem for Brown-York mass. This provides us a further understanding of this conjecture. We will also discuss some closely related topics. This talk is based on a joint work with Wei Yuan. 2018年11月30日10:00–12:00 蒙民伟楼1105室 徐斌(徐州师范大学) Title: On  Ambrosetti-Malchiodi-Ni  conjecture  on  two-dimensional smooth  bounded  domains Abstract: We consider a singularly perturbed elliptic problem on a smooth two dimensional bounded domain. Let $\Gamma$ be a curve intersecting orthogonally with the boundary at exactly two points and dividing the domain into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the arc length functional . We prove the existence of a solution concentrating along the whole of $\Gamma$, exponentially small at any positive distance from it, provided that small parameter is small and away from certain critical numbers.  In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni ( Indiana Univ. Math. J 53(2), 297-329, 2004). This is a joint work with Suting Wei,  Jun,Yang. 2018年12月7日10:00–12:00 蒙民伟楼1105室 孙俊(武汉大学) Title: Invariant Submanifolds of Kahler Manifold Abstract: This talk consists of two parts related to the invariant submanifold of Kahler manifold. The first part is about the variational characterizations of holomorphic submanifolds in Kahler manifold. This is based on joint work with Claudio Arezzo at ICTP. We will also mention recent progess on variational characterizations of invariant submanifold in Sasaki manifold. The second part is about two approaches to the existence of holomorphic curves in Kahler surface. One is the symplectic mean curvature flow and the other is the deformation of symplectic critical surfaces. This is based on joint work with Xiaoli Han and Jiayu Li. 2018年12月14日 10:00–12:00 蒙民伟楼1105室 沈伟明(首都师范大学) Title: On the global properties of the singular Yamabe metrics  Abstract: A version of the singular Yamabe problem in bounded domains in Riemannian manifolds yields complete conformal metrics with negative constant scalar curvature. In this talk, I will talk about some global properties of these singular Yamabe metrics. We will first discuss whether the singular Yamabe metrics have negative Ricci curvatures or sectional curvatures. Then we will talk about the properties of the first global terminal the polyhomogeneous expansions for the singular Yamabe metric in dimension 2. 2018年12月21日10:00–12:00 蒙民伟楼1105室 吴鹏(复旦大学) Title: Gradient shrinking Ricci solitons of half harmonic Weyl curvature Abstract: Gradient Ricci solitons are natural generalizations of Einstein manifolds, and they play important roles in the study of the Ricci flow. In this talk I will talk about the classification of radient shrinking Ricci solitons with half harmonic Weyl curvature. The basic idea of the proof, which is motivated by the study of Einstein four-manifolds, is to construct a weighted subharmonic harmonic function using the Weitzenbock formula for the half Weyl curvature and apply a weighted maximum principle. This is joint work with Jia-Yong Wu and William Wylie. 2018年12月28日10:00–12:00 蒙民伟楼1105室 陆思远(Rutgers Univ.) Title: On a localized Riemannian Penrose inequality Abstract: For a bounded manifold with nonnegative scalar curvature, the Brown-York quasi-local mass is nonnegative and equals to 0 iff it's a domain in Euclidean space by fundamental results of Shi and Tam. Moreover, it is shown that the inequality is equivalent to Positive mass theorem. We consider the general setting that the bounded manifold allows a horizon. We establish a localized Riemannian Penrose inequality and prove that the equality holds iff it's a domain in Schwarzschild manifold. Similar to the Shi-Tam case, the inequality is equivalent to Riemannian Penrose inequality. This is based on joint works with P. Miao. 2019年1月11日9:00–10:30 蒙民伟楼1105室 傅鑫(Rutgers Univ.) Title: On local Kahler-Einstein metric near isolated singularity Abstract: We report some recent results on construction of local Kahler-Einstein metrics and their geometry. 2018年1月11日10:30–12:00 蒙民伟楼1105室 王文龙(北京大学) Title: Rigidity of Riemannian Penrose inequality for asymptotically flat 3-manifolds with corners Abstract: In this talk, we will talk about a rigidity result for the equality case of the Penrose inequality on asymptotically flat 3-manifolds with nonnegative scalar curvature and corners. This result is closely related to the boundary behavior of compact manifolds with nonnegative scalar curvature. This is joint work with Prof. Yuguang Shi and Dr. Haobin Yu.