Flow by Gauss curvature to the $ L_p $ dual Minkowski problem

Abstract

In the paper [20], the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper [20] treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $ L_p $ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from [13] to find a special initial condition such that the Gauss curvature flow converges to a solution of the $ L_p $ dual Minkowski problem.

Keywords

• gauss curvature flow
• monge-ampere equation
• minkowski problem
• variational method
• a priori estimates