A logarithmic Gauss curvature flow and the Minkowski problem

✍ Scribed by Kai-Seng Chou; Xu-Jia Wang

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 125 KB
• Volume: 17
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(00)00053-6

✦ Synopsis

Let X 0 be a smooth uniformly convex hypersurface and f a postive smooth function in S n . We study the motion of convex hypersurfaces X(•, t) with initial X(•, 0) = θX 0 along its inner normal at a rate equal to log(K/f ) where K is the Gauss curvature of X(•, t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ * > 0 such that if θ < θ * , they shrink to a point in finite time and, if θ > θ * , they expand to an asymptotic sphere. Finally, when θ = θ * , they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f (x).