微分几何与几何分析讨论班(2023春季学期)

时 间

地 点

报告人

题目与摘要

 2023年3月14日10:00-12:00

蒙民伟楼1105室 

夏超(厦门大学)

Title: Alexandrov’s theorem for anisotropic capillary hypersurfaces in half-space

Abstract: The minimizers for surface free energy functional, which is the sum of anisotropic surface tension (or parametric elliptic functional) and the wetting energy functional, in half-space are known to be truncated Wulff shapes. The anisotropic capillary hypersurfaces arise as the critical points of the free energy functional under volume constraint.

In this talk, we prove an Alexandrov-type theorem saying that any embedded anisotropic capillary hypersurfaces in half-space are truncated Wulff shapes. The main ingredients are Heintze-Karcher-type inequality and Minkowski-type formula.

The talk is based on a joint work with Xiaohan Jia, Guofang Wang and Xuwen Zhang.

2023年3月21日10:00—12:00

蒙民伟楼1105室

马世光(南开大学)

Title: Conformal metrics with prescribed curvature functions

Abstract: The Scalar curvature can be regarded as the trace of Schouten tensor, and is related to Yamabe equation. As a generalization, we consider the fully nonlinear equation of prescribed sigma_k function of Schouten tensor, with Dirichlet boundary condition.  I will talk about the history of this problem, the basic approach to proving the existence of the solutions and the recent progress we are making.

2023年3月24日11:00–12:00

蒙民伟楼1105室 

莫小欢(北京大学)

Title: Finsler warped product metrics

Abstract: In this lecture we discuss the warped structures of Finsler metrics. We obtain the differential equation that characterizes the Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics. We also discuss a class of Finsler warped product metrics with quadratic Weyl curvature. We give necessary and sufficient conditions of such metrics to be of quadratic Weyl curvature which are non-trivial in the sense that these metrics are not of Weyl type, refining a theorem due to Gabrani-Sevim-Shen.  As its application, we construct infinitely many new non-trivial W-quadratic Finsler warped product metrics.

2023年3月28日

10:00—12:00

西大楼210室 

陈世炳(中国科技大学)

Title: 最优部分传输理论

Abstract: 最优部分传输（Optimal Partial Transport）作为最优传输理论的一个重要分支，关注在给定质量传输约束下如何以最小成本将质量从一个地点迁移到另一个地点。与最优传输问题的不同之处在于，最优部分传输涉及的是传输源和目标分布中的部分质量，而非全部质量。最优部分传输问题在计算机图形学、图像处理、机器学习等领域具有广泛的实际应用价值。从分析学角度出发，研究这个问题中涉及的自由边界的正则性问题具有巨大的挑战性。在本次报告中，我们将重点介绍近年来在这一领域取得的相关成果和研究进展。

2023年4月11日10:00—12:00

蒙民伟楼1105室 

王友德(中国科学院数学与系统科学研究院,、广州大学)

Title: 薛定谔流的诺依曼边值问题  

Abstract: 我们将回顾作为薛定谔流的物理背景的Landau-Lifshitz方程及其相关方程的历史，及其此方程与物理学（微电子学）、材料科学、流体力学的紧密联系。另一方面，也回顾此类方程与微分几何与拓扑学之间自然的联系。最后，介绍我们最近就薛定谔流（Landau-Lifshitz方程）的初始-诺依曼边值问题的强解及光滑解的存在性所取得的进展。 

2023年4月13日10:00—12:00  (周四)

戊己庚楼203-3室 

史宇光(北京大学)

Title: Rigidity  and  non-rigidity of  $H^n/Z^{n-2}$ with  scalar  curvature  bounded  from  below 

Abstract: We present a counterexample to a generalization of Min-OO's hyperbolic rigidity theorem proposed by M.Gromov, and also prove a rigidity result of ALH manifolds with scalar curvature bounded from below. This talk is based on my recent joint work with my postdoc Y.H. Hu and my Ph.d. students P.Liu, T. Z.Hao. 

2023年4月25日10:00—12:00

蒙民伟楼1105室 

刘佳伟(南京理工大学)

Title: The conical Kähler-Ricci flow and its related topics

Abstract: In this talk, I will first talk about the existence, regularity and uniqueness of the conical Kähler-Ricci flow on compact Kähler manifold, and then the stability of this flow on Fano manifold and its applications. At last, I will talk about the related open problems.

2023年5月5日9:00—11:00

西大楼108室 

江文帅(浙江大学)

（Cancelled, TBA）

Title: Critical sets of solution to elliptic equations with weak regular coefficients

Abstract: In this talk, we will consider elliptic equation of divergence form with Holder coefficient of the leading term.  We will consider the Hausdorff measure and Minkowski content estimates of the critical sets of the solutions. By elliptic theory, we know that the Holder coefficient condition is the weakest condition to make sure the solution to be C^1 so that the critical set could be well defined. If the coefficient is only C^0, the solution in general is only W^{1,p} or C^\alpha for any p<\infty, 0<\alpha<1. This is a joint work with Yiqi Huang from MIT.

2023年5月9日10:00–12:00

蒙民伟楼1105室

欧剑宇(厦门大学)

Title: Liouville properties on gradient shrinking Ricci solitons with constant scalar curvature

Abstract:  In this talk we show that bounded harmonic functions are constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension. This talk is based on the joint work with Weixiong Mai.

2023年5月16日10:00—12:00

蒙民伟楼1105室 

王芳(上海交通大学)

Title: Some recent progress on fractional GJMS operators

Abstract: In this talk, I will first give some pointwise comparison formulae for fractional Q-curvatures of different orders. These imply the rigidity theorem for Poincare-Einstein manifold and the comparison theorem for Green functions of fractional GJMS operators. This is joint work with Huihuang Zhou.

2023年5月18日10:00–12:00 (周四)

戊己庚楼203-3室 

朱锦天(北京大学)

Title: Gauss-Bonnet inequality beyond the aspherical conjecture

Abstract: In this talk, I'll make a discussion on recent progress in the research of scalar curvature. I'll first review the backgrounds on the aspherical conjecture, and then focus on the following generalized Gauss-Bonnet inequality: if the universal covering of a closed manifold with nonnegative scalar curvature has ``homological dimension no greater than two'', then either it is flat or its Gauss-Bonnet quantity is no greater than $8\pi$, where the Gauss-Bonnet quantity is the infimum of ambient scalar curvature integral over non-contractible two-spheres.

2023年5月23日 10:00–12:00

蒙民伟楼1105室   

殷浩(中国科技大学)

Title: On the blow-up of Yang-Mills connection in dimension four

Abstract: We study the blow-up of a sequence of Yang-Mills connection with bounded energy on a four manifold. We prove a set of equations relating the geometry of the bubble connection at the infinity with the geometry of the limit connection at the energy concentration point.These equations exclude certain scenarios from happening. The proof involves the expansion of connection forms with respect to some Coulomb gauge on long cylinders.

2023年5月25日 9:00–10:30

戊己庚楼203-3室  

张希(南京理工大学)

Title:  The Hermitian-Yang-Mills equation and its applications

Abstract:  In this talk, we first review the Donaldson-Uhlenbeck-Yau theorem about the solvability of the Hermitian-Yang-Mills equation. We consider mean curvature positivity of holomorphic vector bundle.  By using the perturbed Hermitian-Yang-Mills equation, we will show that that the mean curvature positivity is equivalent to the HN-positivity. As its applications, We establish the correspondence between rational connectedness in algebraic geometry and mean curvature positivity in differential geometry. This work is joint with Chao Li and  Chuanjing Zhang.

2023年5月25日 10:30–12:00

戊己庚楼203-3室  

周斌(北京大学)

Title:  Singular Abreu equations and linearized Monge-Ampère equations with drifts

Abstract: We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher dimensional case by transforming singular Abreu equations into linearized Monge-Ampère equations with drifts. We establish global Hölder estimates for the linearized Monge-Ampère equation with drifts under suitable hypotheses, and then use them to the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side will also be discussed.

2023年5月30日10:00–12:00 

蒙民伟楼1105室 

丁琪(复旦大学)

Title: Special Lagrangian equations

Abstract: In this talk, I will discuss the rigidity and regularity of special Lagrangian equations on Euclidean space, as well as some related existence results. 

2023年6月15日10:00–12:00 

戊己庚楼203-3室 

陆思远 (McMaster University)

Title: Mobius invariant equations in dimension

Abstract:  Conformal invariant equations in n\geq3 have played an important role in the study of \sigma_k-Yamabe problem in geometric analysis. In this talk, we will discuss a class of Mobius invariant equations in dimension two and then present a Liouville type theorem for such equations. We will then discuss the \sigma_2-Nirenberg problem on S^2. This is based on joint works with YanYan Li and Han Lu.

2023年6月 日 10:00–12:00

蒙民伟楼1105室 

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2023年6月 日10:00–12:00 

蒙民伟楼1105室 

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2023年6月 日10:00–12:00

蒙民伟楼1105室

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2023年7月4日10:00–12:00

蒙民伟楼1105室

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