Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

The evolution of complete non-compact graphs by powers of Gauss curvature

We prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.