# 微分几何与几何分析讨论班(2017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 2017年9月12日10:15–12:15 蒙民伟楼1105室 朱苗苗(上海交通大学) Title: Boundary Value Problems for Dirac-harmonic Maps and Their Heat Flows Abstract: In this talk, we shall discuss some boundary value problems for Dirac-harmonic maps and present some recent progresses on the heat flow approach to the existence problem. 2017年9月19日10:15–12:15 蒙民伟楼1105室 王克磊(武汉大学) Title: Finite Morse Index Solutions of the Allen-Cahn Equation   Abstract: In this talk I will report a recent proof of the conjecture that finite Morse index solutions of the Allen-Cahn equation in the plane have finitely many ends. I will discuss some ideas in this proof, including the curvature estimate of Schoen type, the uniform second order estimates of clustering interfaces and its connection with Toda systems. This is a joint work with Juncheng Wei. 2017年9月26日10:15–12:15 蒙民伟楼1105室 丁琪(复旦大学) Title: Minimal hypersurfaces in manifolds with nonnegative curvature Abstract: In this talk, I would like to introduce the theory of minimal hypersurfaces in manifolds with nonnegative Ricci curvature. On the one hand, I will review the classic results on minimal surfaces in 3-dimensional manifolds, then talk about minimal hypersurfaces in high dimensional manifolds with nonnegative Ricci curvature. On the other hand, I will talk about minimal graphs in product manifolds $\Sigma\times \mathbb{R}$, where $\Sigma$ has nonnegative Ricci curvature. In particular, I will focus on gradient estimates, splitting theorem, Liouville type theorem as well as non-trivial examples of minimal graphs. 2017年10月3日 NO NO NO 2017年10月8日10:00—12:00 蒙民伟楼1105室 王国芳(Freiburg Univ.) Title: Uniqueness of stable free boundary CMC hypersurfaces in a ball Abstract: In this talk we will present a solution of a longstanding open problem: Any stable free boundary CMC hypersurfaces in a ball are umbilic. One of crucial ideas is a new weighted Minkowski identity for free boundary hypersurfaces. Our proof works also for capillary hypersurfaces in a ball in a space form. This is a joint work with Chao Xia (Xiamen University). 2017年10月10日10:15—12:15 蒙民伟楼1105室 麻希南(中国科技大学) Title: 非线性椭圆偏微分方程的Neumann问题 2017年10月17日10:15—12:15 蒙民伟楼1105室 刘保平(北京大学BICMR) Title: Stable soliton resolution for exterior wave map in 3d Abstract: Dissipation of energy by dispersion is the key mechanism of relaxation to a static equilibrium in infinite dimensional Hamiltonian systems on unbounded domains. In mathematical language, this is described as the soliton resolution conjecture.  Despite its great importance, the rigorous study is still at a very early stage. In this talk we consider the equivariant wave map exterior to a ball in R^3 and takes values in 3-sphere. We prove that an arbitrary l-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. This talk is based on joint works with Carlos Kenig, Andrew Lawrie and Wilhelm Schlag. 2017年10月24日10:15—12:15 蒙民伟楼1105室 江文帅(浙江大学) Title: Structure of noncollapsing Ricci limit spaces Abstract: Let (M_i^n,g_i,p_i) \to (X,d,p) satisfy Ric>= -(n-1) and Vol(B_1(p_i))>v>0. From Cheeger-Colding theory, for each integer k2$, we also introduce some new result on a$n\$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed  Riemannian manifold. Finally, we mention some applications of the n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopic class. 2017年12月12日10:15--12:15 蒙民伟楼1105室 张晓(中科院数学所) Title: Some mathematical problems in gravitational waves Abstract:On 11 February 2016, the LIGO Scientific Collaboration announced that they detected gravitational waves on 14 September 2015 from a 1.3 billion light years distant merger of two black holes. The nonlinear effects of gravitational waves are described by Bondi-Sachs spacetimes. In the case of zero cosmological constant, they were originally introduced by Bondi for axi-symmetric spacetimes, and generalized by Sachs to general asymptotically flat spacetimes over 50 years ago. Bondi also defined the Bondi energy-momentum at null infinity, which represents the rest energy of spacetimes after the loss due to gravitational radiation. In this talk, we shall discuss some basic problems in gravitational waves for zero cosmological constant. Whether the Bondi energy is nonnegative, i.e., whether the gravitational waves can carry away more energy than they initially have? How the Bondi energy-momentum relates to the ADM energy-momentum defined at spatial infinity? As 1998's cosmological observation indicated our universe has a positive cosmological constant, the Bondi-Sachs spacetimes for positive cosmological constant gain much attention currently. We shall also discuss the natural boundary condition and the surprising peeling property for the Bondi-Sachs spacetimes in this case. 2017年12月19日 蒙民伟楼1105室 NO NO 2017年12月26日 蒙民伟楼1105室 NO NO 2018年1月2日10:15--12:15 蒙民伟楼1105室 孙黎明(The Johns Hopkins Univ.) Title:Convergence of the Yamabe flow on manifolds with minimal boundary Abstract: Analogous to the Yamabe problem, a very natural question on a compact manifold with boundary is deforming Riemannian metrics to conformal ones with constant scalar curvature and  minimal boundary. We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin. This is joint work with Sergio Almaraz. 2018年1月5日10:00—12:00 蒙民伟楼1105室 傅鑫(Rutgers University) Title: Geometric estimate of Monge-Ampere equation Abstract: We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions. 2018年1月8日10:00—12:00 蒙民伟楼1105室 周斌(北京大学) Title: Properness of energy functions on polarized compactifications of reductive Lie groups Abstract: In this talk, I will first give an introduction on Tian's properness conjecture concerning on an analytic characterization of the existence of canonical metrics in Kahler geometry. Then I will focus on compactifications of reductive Lie groups. The main results are criterion theorems of the properness of two important functionals——Ding functional and Mabuchi's K-energy on these manifolds. In particular, the existence of Kahler-Einstein metrics, Kahler-Ricci solitons and Mabuchi's generalized Kahler-Einstein metrics on Fano compactifications of reductive Lie groups can be established. More generally, for constant scalar curvature metrics case, we will obtain the existence of weak minimizers in the sense of convex potentials. 2018年1月16日 蒙民伟楼1105室 NO NO