A logarithmic Gauss curvature flow and the Minkowski problem

  • Kai-Seng Chou

    Department of Mathematic, The Chinese University of Hong Kong, Shatin, Hong Kong
  • Xu-Jia Wang

    School of Mathematical Sciences, Australian National University, Canberra, ACT0200, Australia


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