Winter 2023
May 26, 2023 Fri, 2:00-3:00PM,
Tencent Meeting
Changyu Guo, Shandong University
Geometric parametrization of metric spaces:
In this talk, we shall give an overview of the geometric parametrization problem of metric spaces, which is a very active topic in the recent research. We shall discuss some natural connections with Analysis on metric spaces, Geometric measure theory, Geometric analysis (around harmonic maps) and Harmonic analysis.
May 19, 2023 Fri, 2:00-3:00PM,
Tencent Meeting
Bangxian Han, University of Science and Technology of China
最优传输理论与度量几何:
最优传输理论在过去三十年中得到了巨大发展,在几何, 分析,概率等领域有诸多重要应用. 在这个报告中, 我们将介绍最优传输理论在度量几何研究中的发展历程以及最新应用, 并将简要介绍度量几何是如何影响最优传输理论发展的.
May 05, 2023 Fri, 2:00-3:00PM,
Tencent Meeting
Hengyu Zhou, Chongqing University
The area minimizing problem in conformal cones:
In 1985, Lin Fanghua observed the existence and uniqueness of area-minimizing currents with graphical boundaries in Euclidean cylinders. Motivated by the Plateau problem, it is natural to ask similar problems in the case of conformal cones. In this talk, we report some results on this topic, emphasizing their connections with minimal graphs and prescribed mean curvature equations. Some questions are also presented.
Apr 25, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Zunwu He, South China University Of Technology
On Steklov eigenvalues of trees
In this talk, first, we give upper bounds of all Steklov eigenvalues of finite trees via various geometric quantities: volume of boundary, diameter, etc. Second, by introducing Steklov flows, we show the monotonicity of the first Steklov eigenvalue and give a complete characterization of the rigidity. These are joint works with Bobo Hua.
Apr 18, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Bin Deng, Wuhan University
Quantitative stability of harmonic maps from $R^2$ to $S^2$ with a higher degree:
For degree ±1 harmonic maps from $R^2$ (or $S^2$ ) to $S^2$, Bernand-Mantel, Muratov and Simon established a uniformly quantitative stability estimate. Namely, for any map $u:R^2\to S^2$ with degree ±1, the discrepancy of its Dirichlet energy and $4\pi$ can linearly control the $H^1$-difference of $u$ from the set of degree ±1 harmonic maps. Whether a similar estimate holds for harmonic maps with a higher degree is unknown. We prove that a similar quantitative stability result for a higher degree is true only in a local sense. Namely, given a harmonic map, a similar estimate holds if is already sufficiently near to it (modulo M\"{o}bius transforms) and the bound in general depends on the given harmonic map. More importantly, we thoroughly investigate an example of the degree 2 case, which shows that it fails to have a uniformly quantitative estimate like the degree ±1 case. This phenomenon shows the striking difference between degree ±1 ones and higher degree ones.Joint with L.M Sun & J.C. Wei.
Apr 11, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Liding Huang, Westlake University
The Cauchy-Dirichlet problem for parabolic deformed Hermitian-Yang-Mills equation
We will talk about the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $\Omega\subset C^{n}$. By using J-functional, we can prove the convergence of solutions. As an application, we will give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
Apr 04, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Guodong Wei, Sun Yat-sen University (Zhuhai)
On the fill-ins with scalar curvature bounded from blow:
A triple of (generalized) Bartnik data $(\Sigma^{n-1},\gamma,H) $ consists of an orientable closed null-cobordant Riemannian manifold $(\Sigma^{n-1},\gamma)$ and a given smooth function $H$ on $\Sigma^{n-1}$. One basic problem in Riemannian geometry is to study : it under what kind of conditions does the Bartnik data $(\Sigma^{n-1},\gamma,H) $ admit a fill-in metric $g$ with scalar curvature bounded below by a given constant? In this talk, we will discuss some estimates on the mean curvature of fill-ins with scalar curvature bounded from blow. This is mainly based on joint work with Wenlong Wang at Nankai University.
Mar 28, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Jinyu Guo, Tsinghua University
Stable capillary hypersurfaces supported on a horosphere in the hyperbolic space:
In this talk, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.This is a joint work with Prof.Guofang Wang and Prof.Chao Xia.
Mar 21, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Xiaoli Han, Tsinghua University
Existence of mean curvature flow singularities:
Velazquez constructed a countable collection of mean curvature flow solutions in R^N, N\geq 8. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. Guo and Sesum , Stolarski studied the behavior of the mean curvature of Velazquez's solution. While the construction provided by Velazquez's yields complete, non-compact mean curvature flow solutions. We will construct closed mean curvature flow solutions which become singular in finite time.
Mar 14, 2023 Tue, 9:00-10:00AM,
Tencent Meeting
Yan Xu, Nankai University
Classification of constantly curved holomorphic 2-spheres of degree 6 in the complex Grassmannian G(2,5):
Up to now the only known example in the literature of constantly curved holomorphic 2-sphere of degree 6 in the complex G(2, 5) has been the first associated curve of the Veronese curve of degree 4. By exploring the rich interplay between the Riemann sphere and projectively equivalent Fano 3-folds of index 2 and degree 5, we prove, up to the ambient unitary equivalence, that the moduli space of generic (to be precisely defined) such 2-spheres is semi-algebraic of dimension 2. All these 2-spheres are verified to have non-parallel second fundamental form except for the above known example. This is a joint work with Professor Q.S. Chi and Z.X. Xie.