A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

TL;DR

This paper investigates anisotropic inverse Gauss curvature flows, establishing long-term behavior and deriving new existence results for dual Orlicz Minkowski problems, extending classical $L^p$ problems.

Contribution

It introduces a novel class of anisotropic inverse Gauss curvature flows and connects their stationary solutions to new existence results for dual Orlicz Minkowski problems.

Findings

Established long-time existence and asymptotic behavior of anisotropic inverse Gauss curvature flows.

Derived new existence results for dual Orlicz Minkowski problems for smooth measures.

Extended the $L^p$ dual Minkowski problem to broader parameter ranges.

Abstract

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<q$, which contain all existence results for smooth measures up to now except $p=q$ or $q=n+1$ ($L^p$ Minkowski problem).