微分几何与几何分析讨论班(2017秋季学期)

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 k<n, we know that the stratification S^k={ x\in X: no tangent cone at x splits an R^{k+1} factor }  has dimension =<k. In this talk, we will show that S^k is k-rectifiable. We will also discuss the quantitative measure estimate of  S^k and some applications.   This is joint work with J.Cheeger and A. Naber.

2017年10月31日10:15—12:15

蒙民伟楼1105室 

王雪平(Univ. of Nantes)

Title: Gevrey estimates of the resolvent and sub-exponential time-decay of solutions

Abstract: Spectral analysis of non-selfadjoint Schrödinger operators is a topic related to several branches of mathematics. In this talk, we study a 

class of non-selfadjoint Schrödinger operators which are compactly supported perturbation of some model operator verifying a weighted coercive condition. We establish large-time expansion of solutions to the time-dependent Schrödinger equation with sub-exponential time-decay estimate on the remainder, including the case of zero eigenvalue and positive resonances.

2017年11月7日 14:00—16:00

蒙民伟楼1105室 

周恒宇(中山大学)

Title: Some curvature flows in warped product manifolds

Abstract: In this talk, we will discuss some curvature flows in warped product manifolds.  They include curve shortening flows, mean curvature type flows and inverse mean curvature flows.  If the initial datum is a starshaped graph,  we will see how various conditions imply long time existence and certain asymptotic behaviors of these flows.  We also discuss the connection between our results and some other recent works.

2017年11月14日10:15--12:15 

蒙民伟楼1105室 

王枫(浙江大学)

 Title: The existence of Kahler-Einstein metrics on K-polystable Q-Fano varieties with non-positive  discrepancies

Abstract: I will talk about the recent work with Professor Tian and Chi Li. At first, we extend Tian’s work to the log smooth case. Then for a K-polystable Q-Fano varieties X with non-positive  discrepancies, we show that there exists conic KE metrics on the resolution and these metrics converges to the singular KE metric on X.

2017年11月21日10:15--12:15 

蒙民伟楼1105室 

朱小华(北京大学)

Title:Steady Ricci solitons with positive curvature

Abstract: I will talk about our recent  results related to the rigidity of Steady Ricci solitons in higher

dimension. We prove that any  noncompact $\kappa$-

noncollapsed steady Ricci soliton with nonnegative curvature operator and positive Ricci curvature must be rotationally symmetric if it  has a  linear curvature decay. If a $\kappa$-noncollapsed steady Ricci soliton is Kaehler with nonnegative bisectional curvature, then it is flat.

2017年11月28日 

蒙民伟楼1105室 

2017年12月5日10:15--12:15 

蒙民伟楼1105室 

洪敏纯(Univ. of Queensland)

Title: The heat flow for harmonic maps and applications

Abstract:In 1964, Eells and Sampson asked whether  a given smooth map   can be deformed to a harmonic map  in its homotopy class . When $n=2$, Lemaire   and Schoen-Yau established  existence results of harmonic maps by minimizing the Dirichet energy  in a homotopic class under the  topological condition $\pi_2(N)=0$.   Sacks and Uhlenbeck established many existence results  of minimizing harmonic maps in their homotopic classes by introducing the `Sacks-Uhlenbeck functional'. To solve the  Eells-Sampson question,  it is important to establish  global existence of a solution of the harmonic map flow. Struwe proved  the existence of a unique global weak solution  to the harmonic map flow.   Chang, Ding and Ye constructed an example that the harmonic map flow  blows up at finite time.  Ding and Tian established the energy identity of the harmonic map flow at each blow-up time.

When $n>2$, 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