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

Abstract

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 $L_p$ Gauss image problem

“…The L 0 Gauss image problem is just the Gauss image problem which was first mentioned in [5], and the existence of smooth solution for the Gauss image problem was in [14]. It is necessary to contrast the L p Gauss image problem with the various Minkowski problems and dual Minkowski problems that have been extensively studied, see [8,11,15,16,27,29,30,34,35,36,37,38,40,44,45] for the L p -Minkowski problem, [7,21,23,24,32,42,43] for the dual Minkowski problem, [6,9,10,25,26,31,39] for the L p dual Minkowski problem, [4,20,22,28] for the Orlicz Minkowski problem, [12,13,18,19,33] for the dual Orlicz Minkowski problem, [17] for the Orlicz Aleskandrov problem.…”