α-Gauss Curvature flows with flat sides

Lami Kim; Ki-ahm Lee; Eunjai Rhee

paper

α-Gauss Curvature flows with flat sides

arXiv:1109.6699

Abstract

In this paper, we study the deformation of the 2 dimensional convex surfaces in $R^{3}$ whose speed at a point on the surface is proportional to $α$ -power of positive part of Gauss Curvature. First, for 1/2<α\leq 1 $, w es h o w t ha tt h er e i ss m oo t h so l u t i o ni f t h e ini t ia l d a t ai ss m oo t han d s t r i c tl y co n v e x an d t ha tt h er e i s a v i scos i t y so l u t i o n w i t h$ C^{1,1} $- es t ima t e b e f or e t h eco l l a p s in g t im e i f t h e ini t ia l s u r f a ce i so n l y co n v e x . M or eo v er, w es h o w t ha tt h er e i s a w ai t in g t im ee f f ec tw hi c hm e an s t h e f l a t s p o t o f t h eco n v e x s u r f a ce w i l l p er s i s t f or a w hi l e . W e a l sos h o w t h e in t er f a ce b e tw ee n t h e f l a t s i d e an d t h es t r i c tl y co n v e x s i d eo f t h es u r f a cer e main ss m oo t h o n$ 0 < t < T_0 $u n d er cer t ainn ecess a r y r e g u l a r i t y an d n o n - d e g e n er a cy ini t ia l co n d i t i o n s, w h er e$ T_0$ is the vanishing time of the flat side.