微分几何与几何分析讨论班(2019春季学期)

时间：周五上午（或周六下午）

地点：蒙民伟楼1105报告厅

时间	地点	报告人	题目与摘要
2019年3月22日10:00–12:00	蒙民伟楼1105室	周斌(北京大学)	Title: A four vertex theorem for space curve on locally convex surface Abstract: The classical four vertex theorem characterizes an interesting property of closed planar curves. It has been extended to space curves on sphere and on more general convex surfaces, namely a smooth, closed curve in R^3 has at least four points with vanishing torsion if it lies on a convex surface. In this talk we discuss the four vertex theorem for space curves on locally convex surfaces by using the properties of homogeneous Monge-Ampere equation. This is a joint work with Shibing Chen and Xu-jia Wang.
2019年3月29日10:00-12:00	蒙民伟楼1105室	张立群(中科院数学所)	Title: On the regularity of weak solutions of parabolic equations Abstract: We review the proof of regularity of weak solutions for heat equations and then consider a class of ultraparabolic differential equations with measurable coefficients and lower order terms. It is proved that weak solutions to the equation are Holder continuous, which generalizes the classic results of parabolic equations with general coefficients. The main observation is a weak Sobolev inequality and Poincare inequality related to weak sub-solutions. This is a jointed work with Wang Wendong.
2019年4月12日10:00—12:00	蒙民伟楼1105室	周家足(西南大学)	Title: L_(p,q) curvature measures and L_(p,q) Minkowski problems in integral and convex geometry Abstract: We will introduce Lp dual curvature measures (L_(p,q) curvature measures) defined recently by Lutwak-Yang-Zhang and address some recent results on L_(p,q) Minkowski problems in integral and convex geometry.
2019年4月18日15:00–17:00	西大楼210室	韩飞(National University of Singapore)	Title: Modular invariants for proper actions Abstract: In this talk, we will present our work on open Lie group actions on open manifolds. Witten genus and elliptic genera are modular topological invariants for manifolds, which are closely related to representation of loop groups and the hypothetical index theory on free loop space as well as the elliptic cohomology theory in algebraic topology. They find applications in problems of positive curvature and group actions on manifolds. In this talk, we will briefly introduce these invariants and present our joint work with Varghese Mathai on generalizing them to open manifolds with proper actions of open Lie groups.
2019年4月19日10:00—12:00	蒙民伟楼1105室	邓宇星(北京理工大学)	Title: Rigidity of positively curved steady Ricci solitons Abstract: In this talk, we will talk about the rigidity of positively curved steady Ricci solitons. In the Riemannian case, we will talk about the classification of steady Ricci solitons with linear decay. In the Kahler case, we will talk about the rigidity of steady Kahler-Ricci solitons with nonnegative bisectional curvature and its application. It is a joint work with Prof. Xiaohua Zhu.
2019年4月26日10:00—12:00	蒙民伟楼1105室	熊金钢(北京师范大学)	Title: On asymptotic behavior of solutions of conformally invariant equations with isolated singularities Abstract: I will review some classical results of conformally invariant equations with isolated singularities. I will report a recent joint work with Tianling Jin, in which we solved the higher order case. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension.
2019年4月27日14:00–16:00	蒙民伟楼1105室	盛利(四川大学)	Title: A Bernstein Theorem and A Liouville Theorem Abstract: In this talk we give an introduction to affine geometry, and consider a class of fourth order PDE and establish a Bernstein Theorem by affine techniques. Next we talk a Liouville Theorem on the PDE det(D^2 f) = 1 under some assumption.
2019年5月5日10:00—12:00	蒙民伟楼1105室	张若冰(Stony Brook Univ.)	Title: Geometric analysis of collapsed Einstein spaces Abstract: This talk centers on the geometric analysis of a family of collapsing Einstein manifolds with sufficiently wild analytic properties, for instance, the uniform Sobolev inequality never holds in any collapsing sequence. We will introduce some new techniques from both metric-geometric and algebra-geometric sides in analyzing Einstein equations. A specific topic is that, for any arbitrary dimension, we have managed to construct a large variety of collapsed Ricci-flat Kaehler spaces with degenerating complex structures, which is for the first time in the literature.
2019年5月10日10:00—12:00	蒙民伟楼1105室	NO	NO
2019年5月17日10:00—12:00	蒙民伟楼1105室	NO	NO
2019年5月24日 10:00–12:00	蒙民伟楼1105室	韩小利(清华大学)	Title: On line bundle mean curvature flows Abstract: We will introduce the definition of deformed Hermitian-Yang-Mills metric and the line bundle mean curvature flows which is defined by Jacob-Yau. Then we will provide a monotonicity formula along the line bundle mean curvature flow. Using this monotone quantity we define the Gaussian density. Under the assumption of the Gaussian density, we will give an \epsilon-regularity theorem.
2019年5月31日10:00–12:00	蒙民伟楼1105室	夏超(厦门大学)	Title: New Minkowski type formulas for free boundary hypersurfaces in balls and applications Abstract: In this talk, we will present a class of new Minkowski formulas for free boundary hypersurfaces, or more generally capillary hypersurfaces, in balls. Several applications will be given. First, we use it to classify all stable capillary hypersurfaces in balls to be umbilical ones. Second, we use the free boundary version to prove an Alexandrov type theorem for embedded free boundary CMC hypersurfaces in half balls. Third, we define a class of locally constraint inverse type curvature flows and show a family of Alexandrov-Fenchel’s inequalities for free boundary hypersurfaces in balls. The talk based on joint works with Guofang Wang and Julian Scheuer.
2019年6月8日14:00–16:00	蒙民伟楼1105室	殷浩(中国科技大学)	Title: Higher order neck analysis for harmonic maps and its applications Abstract: We shall prove some refined estimate on the neck region when a sequence of harmonic maps from surfaces blow up. It generalizes the well-known energy identity and no neck theorem of harmonic maps. We then discuss applications, which include a further blow-up of the neck region and an inequality about the nullity and index of the sequence.
2019年6月14日 10:00–12:00	蒙民伟楼1105室	来米加(上海交通大学)	Title: Hang-Wang type Rigidity theorem Abstract: Let (M, g) be a compact Riemannian manifold with boundary. Suppose Ric \geq n-1, Hang and Wang proved that M is isometric to the standard hemisphere provided that $\partial M$ is isometric to $S^{n-1}$ and convex. In this talk, we present a generalization of this result. We assume the $\partial M $ is isometric to a product manifold with one sphere factor, and pose some conditions on the second fundamental form of $\partial M$. It turns out $M$ is isometric to a doubly warped product metric.
2019年6月21日10:00–12:00	蒙民伟楼1105室	杨瑞杰(Stony Brook Univ.)	Title: Extensions of holomorphic forms Abstract: Given a family of n-dimensional complex projective manifolds over a base B, the spaces of top holomorphic differential forms (i.e. (n,0) forms) on fibers give rise to a holomorphic vector bundle H on B which is equipped with a positive metric. Ohsawa-Takegoshi theorem with sharp estimates, recently proved by Blocki and Guan-Zhou, implies that one can extend top forms on the central fiber with good L2 bounds. This analytic imput only guarantees the existence of extensions. I would like to discuss a project in progress about geometric approaches to this problem for certain families. The main tool is Griffiths’ theory of variation of Hodge structures and the corresponding Degline’s representation-theoretic interpretation. The necessary background will be reviewed in the first part of the talk.
2019年6月28日9:00–10:30	蒙民伟楼1105室	王邵东(上海交通大学)	Title: A compactness theorem for boundary Yamabe problem in the scalar-flat case Abstract: In this talk, I will present some recent results on the compactness of the solutions to the Yamabe problem on manifolds with boundary. The compactness of Yamabe problem was introduced by Schoen in 1988. There have been a lot of works on this topic later on. This is a joint work with Sergio Almaraz and Olivaine Queiroz.
2019年6月28日10:30–12:00	蒙民伟楼1105室	王芳(上海交通大学)	Title: Abstract:
2019年7月1日10:00–12:00	蒙民伟楼1105室	Qi S. Zhang(UC Riverside)	Title: A few properties of global solutions of the heat equation on Euclidean space and some manifolds Abstract: We report some recent results on Martin type representation formulas for ancient solutions of the heat equation and dimension estimates of the space of these solutions under some growth assumptions. We will also present a new observation on the time analyticity of solutions of the heat equation under natural growth conditions. One application is a "iff" solvability condition of the backward heat equation, i.e. under what condition can one turn back the clock in a diffusion process. Part of the results are joint work with Fanghua Lin and Hongjie Dong.