Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

TL;DR

This paper investigates anisotropic curvature flows without global terms to establish existence results for dual Orlicz Christoffel-Minkowski problems, connecting geometric PDEs with convex body theory and extending previous Minkowski problem solutions.

Contribution

It introduces new existence results for dual Orlicz Minkowski problems via anisotropic flows, covering many known cases and addressing a broader class of curvature problems.

Findings

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

Derived inequalities involving modified quermassintegrals.

Provided partial solutions to the prescribed curvature problem.

Abstract

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,Du_K)F(D^2u_K+u_KI)=1$ on $\mathbb S^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on $\mathbb S^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).