几何分析讨论班—FLOW BY GAUSS CURVATURE TO THE ALEKSANDROV AND DUAL MINKOWSKI PROBLEMS
主 题: 几何分析讨论班—FLOW BY GAUSS CURVATURE TO THE ALEKSANDROV AND DUAL MINKOWSKI PROBLEMS
报告人: 盛为民 教授 (浙江大学)
时 间: 2017-04-07 10:10-11:10
地 点: 理科1号楼1479
Abstract: In this talk, I will introduce our recent work on Gauss curvature flow with Xu-Jia Wang and Qi-Rui Li. In this work we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\R^{n+1}$ with the speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. We prove that if $\alpha\ge n+1$, the flow exists for all time and converges smoothly after normalization to a hypersurface, which is a sphere if $f\equiv 1$. Our argument provides a new proof for the classical Aleksandrov problem ($\alpha = n+1$) and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang recently, for the case $q<0 alpha="">n+1$). If $\alpha< n+1$, corresponding to the case $q > 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.
