几何分析研讨班

中科院数学与系统科学研究院

数学研究所

学术报告

几何分析研讨班

时  间：2022.07.08（星期五）

地  点：腾讯会议：855-406-449

n  上午9 :30-10 :30

报告人：李宇翔 （清华大学）

题  目：Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature

摘  要：

Let $(\Sigma,g_0)$ be a smooth surface and $g=e^{2u}g_0$. We say $(\Sigma,g)$ is an Alexandrov Surface if $u\in L^1$ and satisfies

where $\mu$ is a Rodan measure. We define $\mu$ to be the Gauss curvature of $g$ and denote it by $\mathbb{K}_g$.

We will show that $u\in W^{1,q}$ for any $q\in[1,2)$ and $g$ induce a metric $d_g$. Then we will prove that when $(\Sigma,g_0)$ is closed  and $g_k=e^{2u_k} g_0 \in\ M(\Sigma,g_0)$,

if $|K_{g_k}|(\Sigma)<C$, and $K_{g_k}$ converges to a measure $\nu$ in the sense of distributions with $\nu^+(\{x\})<2\pi$ for any $x$, then$u_k$ converges weakly to a function $u$ in $W^{1,q}$ for any $1\leq q<2$, $K_g=\mu$ and $d_{g_k,\Sigma}$ converges to $d_{g,\Sigma}$ uniformly where $g=e^{2u}g_0$. This solves an open problem raised in arXiv:2201.03354.

n  上午10 :30-11 :30

报告人：殷浩 （中国科学技术大学）

题  目：On the blow up of Yang-Mills fields in dimension four

摘  要：In this talk, we study the blow up of a sequence of Yang-Mills fields in dimension four. We show that under certain technical assumptions, the bubble curvature at the infinity point and the weak limit curvature at the blow-up point are related by a set of equations.

n  下午14 :30-15 :30

报告人：陈波 （清华大学）

题  目：Local regular solution to the Neumann problem of the Schr\"odinger flow

摘  要：In this talk, we show the existence and uniqueness of short-time very regular solution to the initial-Neumann boundary value problem of the Schr\"odinger flow for maps from a smooth bounded domain in R^3 into S^2 in the scale of Sobolev spaces. We provide a precise description of the compatibility conditions at the boundary for the initial data.

n  下午15 :30-16 :30

报告人：王蕾 （北京大学）

题  目：Hardy-Sobolev inequalities with distance to the boundary weight functions

摘 要：In this talk, we shall introduce some Hardy-Sobolev inequalities, whose weights are distance functions to the boundary. We will discuss about the sharp constants and extremal functions of the inequalities.