An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

TL;DR

This paper introduces a new class of geometric measures called (p,q)-mixed curvature measures and solves a generalized Christoffel-Minkowski problem using an expanding curvature flow, establishing existence and uniqueness of solutions.

Contribution

It generalizes existing Minkowski problems by defining (p,q)-curvature measures and proves existence and uniqueness of solutions via a novel curvature flow approach.

Findings

Established existence of smooth solutions to the (p,q)-Christoffel-Minkowski problem.

Proved the uniqueness of solutions under certain conditions.

Introduced a new class of geometric measures, the (p,q)-mixed curvature measures.

Abstract

The present paper introduces a new class of geometric measures, the k-th (p,q)-mixed curvature measures, and a natural correspondence-(p,q)-Christoffel-Minkowski problem is proposed. The (p,q)-Christoffel-Minkowski problem posed here can be regarded as a natural generalization of the L_p Christoffel-Minkowski problem and Lp dual Minkowski problem. We investigate and arrive at the existence of smooth solution to the (p,q)-Christoffel-Minkowski problem by a type of expanding curvature flow. Furthermore, the uniqueness result of solutions to the (p,q)-Christoffel-Minkowski problem shall be discussed.