A logarithmic Gauss curvature flow and the Minkowski problem
Kai-Seng Chou
Department of Mathematic, The Chinese University of Hong Kong, Shatin, Hong KongXu-Jia Wang
School of Mathematical Sciences, Australian National University, Canberra, ACT0200, Australia
![A logarithmic Gauss curvature flow and the Minkowski problem cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-aihpc-volume-17-issue-6.png&w=3840&q=90)
Abstract
Let be a smooth uniformly convex hypersurface and a postive smooth function in . We study the motion of convex hypersurfaces with initial along its inner normal at a rate equal to where is the Gauss curvature of . We show that the hypersurfaces remain smooth and uniformly convex, and there exists such that if , they shrink to a point in finite time and, if , they expand to an asymptotic sphere. Finally, when , they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on .
Cite this article
Kai-Seng Chou, Xu-Jia Wang, A logarithmic Gauss curvature flow and the Minkowski problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 6, pp. 733–751
DOI 10.1016/S0294-1449(00)00053-6