# 微分几何与几何分析讨论班(2018春季学期)

### 地点：蒙民伟楼1105报告厅

 时 间 地 点 报告人 题目与摘要 2018年3月17日14:00–16:00 蒙民伟楼1105室 楚健春(中科院数学所) Title: Degenerate complex Monge-Ampere equation and its geometric applications Abstract: In this talk, we will derive the real Hessian estimate for solutions of complex Monge-Ampere equations on compact Kahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. We will also discuss some geometric applications. This is a joint work with Professor Valentino Tosatti and Professor Ben Weinkove. 2018年3月21日14:00–16:00 蒙民伟楼1105室 刘磊(Max Planck Institute, German) Title: The qualitative behavior at the free boundary for approximate harmonic maps from surfaces Abstract: Let $\{u_n\}$ be  a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on   a smooth submanifold $K\subset N$ satisfying $\sup_n(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^2(M)})\leq \Lambda,$ where  $\tau(u_n)$ is the tension field of the map $u_n$. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow  with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time. 2018年3月28日14:00–16:00 蒙民伟楼1105室 王志张(复旦大学) Title: Uniqueness of isometric immersions with the same mean curvature Abstract: Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results when the mean curvature condition is replaced by an σ2-curvature condition. 2018年4月4日14:00–16:00 蒙民伟楼1105室 Zhen-Chao Han(Rutgers Univ.) Title: Some remarks on the symmetry  of complete, locally conformally flat metrics on canonical domains of the round sphere with constant $Q$-curvature Abstract: We will report on some results, jointly with Alice Chang and Paul Yang of Princeton University, on the symmetry  of complete, locally conformally flat metrics on canonical domains of the round sphere with constant $Q$-curvature. More specifically \begin{theorem}  Any complete, conformal  metric  $g$ on $\mathbb S^n \setminus \mathbb S^{l}$ for $l\le \frac{n-2}{2}$  satisfying \label{q1} Q_g \equiv 1\; \text{or \ $0$}, and \label{sp} R_g \ge 0, in $\mathbb S^n \setminus \mathbb S^{l}$ has to be symmetric with respect to rotations of $\mathbb S^n$ which leave $\mathbb S^l$ invariant. \end{theorem} This theorem is a corollary of the following \begin{theorem}\label{thm1} Let $g$ be a conformal, complete metric on $\Omega \subsetneqq \mathbb S^n$ such that  \eqref{q1} and \eqref{sp}  hold in $\Omega$. Then for any ball $B\subset \subset \Omega$ in the canonical metric $g_{\mathbb S^n}$, the mean curvature of its boundary $\partial B$ in metric $g$ with respect to its inner normal is nonnegative. \end{theorem} 2018年4月11日14:00—16:00 蒙民伟楼1105室 朱保成(湖北民族学院) Title: On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies Abstract: We will talk about the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measureand a continuous function there exists a convex bodysuch that  is an optimizer of the following optimization problems: $$\inf/sup_{|L|=\omega_n}\{\int_{S^{n-1}}\varphi(h_L)d\mu;L\in \mathcal{R}_0\}.$$ The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on , the existence of a solution is proved for a nonzero finite measure  on unit sphere  which is not concentrated on any hemisphere of . Another part of this paper deals with the p-capacitary Orlicz-Petty bodies. In particular, the existence of the p-capacitary Orlicz-Petty bodies is established and the continuity of the p-capacitary Orlicz-Petty bodies is proved. 2018年4月14日14:00—16:00 蒙民伟楼1105室 马世光(南开大学) Title: n-Laplacian equation and Ricci curvature Abstract: This is a joint work with Professor Jie Qing. In this talk, I will mention two kinds of problems. The first kind is to study noncompact locally conformally flat manifolds with nonnegative Ricci curvature. The second kind is to study noncompact hypersurfaces with nonnegative Ricci curvature of hyperbolic space. Both problems are related to n-Laplacian equation. To study n-Laplacian equation, we use the potential theory. 2018年4月21日14:00—16:00 蒙民伟楼1105室 张振雷(首师大) Title: Relative volume comparison of Ricci flow and its application Abstract: In the talk I will introduce a relative volume comparison of Ricci flow. It is a refinent of Perelman's no local collapsing theorem. Application to the convergence of Kahler-Ricci flow will also be discussed. It is a joint work with Professor Tian. 2018年4月25日14:00—16:00 蒙民伟楼1105室 杨文(武汉物理与数学研究所) Title: On the $SU(n+1)$ Toda system Abstract: In this talk, we will first discuss the local masses of this Toda system with $n\geq2$. By using Pohozaev identity and the monodromy theory from complex ODE, we are able to determine the local masses for the system and it turns out that the local masses are closely related to the element in the Weyl group of the corresponding Lie algebra. Based on this quantization result, the a priori estimate and some existence results of this system are obtained. 2018年5月2日14:00—16:00 蒙民伟楼1105室 邱国寰(McGill Univ.) Title: Interior curvature estimate for scalar curvature equation Abstract: The interior regularity for scalar curvature equation is a longstanding problem in fully nonlinear PDE theory. Joint with Professor Guan, we partially solved this problem under some convexity condition in any dimensions. In Euclidean three-space, I proved the purely interior curvature estimate for scalar curvature equation.  I will talk about my new observation to the scalar curvature equation in dimension three. 2018年5月9日14:00—16:00 蒙民伟楼1105室 Martin Man Chun Li(Chinese University of Hong Kong) Title: Free boundary minimal surfaces - old and new Abstract: Free boundary minimal surfaces are critical points to the area functional for Riemannian manifolds with boundary. In this talk, we will first give a quick overview of some classical study of such objects during last century. After that, we will discuss some recent results centering around the question of existence, regularity, compactness and rigidity. Particular emphasis will be put on the connection with extremal eigenvalue problems, geometric PDEs and min-max theory. Some of these are joint work with A. Fraser, X. Zhou and N. Kapouleas. 2018年5月16日 14:00–16:00 蒙民伟楼1105室 余成杰(汕头大学) Title: Estimate for higher Steklov eigenvalues Abstract: In this talk, we first give a survey on estimates of Steklov eigenvalues. Then, some recent joint works on higher Steklov eigenvalues generalizing the Hersch-Payne-Schiffer’s and Raulot-Savo’s estimates will be presented. 2018年5月23日14:00–16:00 蒙民伟楼1105室 No No 2018年5月30日14:00–16:00 蒙民伟楼1105室 Ernst Kuwert(Freiburg Univ.) Title: The total squared curvature functional for surfaces in manifolds Abstract: We review joint work with A. Mondino and J. Schygulla on the existence of immersed $2$-spheres minimizing the total squared curvature in a compact Riemannian $3$-manifold $M$. In that paper, we assumed that $M$ has strictly positive sectional curvature to bound the area of the minimizing sequence. In the second part of the talk, we present a recent result with V. Bangert. For a sequence of compact immersed surfaces with bounded total squared curvature in a compact Riemannian $n$-manifold, the area is automatically bounded along the sequence unless there is a complete, totally geodesic immersion into $M$. It is known that the second alternative is not generic. 2018年6月6日 14:00–16:00 蒙民伟楼1105室 No No 2018年6月16日14:00–16:00 蒙民伟楼1105室 胥世成(首都师范大学) Title: The stability and uniqueness of N-structures on collapsed manifolds Abstract: The nilpotent Killing structure constructed by Cheeger-Fukaya-Gromov has been a powerful tool in the study of collapsed manifolds with bounded sectional curvature, and draw people's attention in studying collapsed Einstein manifolds. We will talk about the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $m$-manifold of bounded sectional curvature (or bounded Ricci curvature with a positive conjugate radius lower bound) are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $m$), then the underlying $N$-structures are isomorphic or one is embedded into another as a subsheaf. It generalizes Cheeger-Fukaya-Gromov's locally compatibility of pure $N$-structures for one collapsed metric of bounded sectional curvature. As an corollary, we prove that those pure $N$-structures constructed by various smoothing method to a bi-Lipschitz equivalent $\epsilon$-collapsed metric of bounded sectional curvature are determined by the original metric uniquely modulo a diffeomorphism. 2018年6月20日14:00–16:00 蒙民伟楼1105室 张晓(中科院数学所) Title: Boundary value problems for the Dirac operator with geometric applications Abstract:In the talk we discuss the APS and local boundary value problems for the Dirac operator and provide some eigenvalue estimates. As applications, we give a spinorial proof of the Alexandrov theorem as well as the rigidity of PE manifolds. The talk is based on some joint works with Hijazi, Montiel, Daguang Chen and Fang Wang. 2018年6月27日14:00–16:00 蒙民伟楼1105室 Title:  Abstract: 2018年7月4日14:00–16:00 蒙民伟楼1105室 Title:  Abstract: