2017 Theses Doctoral

The Gauss curvature flow : Regularity and Asymptotic Behavior

Choi, Kyeongsu

This thesis contains the author's results on the evolution of convex hypersurfaces by positive powers of the Gauss curvature. We first establish interior estimates for strictly convex solutions by deriving lower bounds for the principal curvatures and upper bounds for the Gauss curvature. We also investigate the optimal regularity of weakly convex translating solutions. The interesting case is when the translator has flat sides. We prove the existence of such translators and show that they are of optimal class C^1,1. Finally, we classify all closed self-similar solutions of the Gauss curvature flow which is closely related to the asymptotic behavior.