A unified flow approach to smooth, even $L_p$-Minkowski problems

TL;DR

This paper develops a unified flow method to address the existence and behavior of solutions to smooth, even $L_p$-Minkowski problems, extending previous curvature flow techniques with new estimates.

Contribution

It introduces a novel flow approach combined with adapted curvature estimates to solve the $L_p$-Minkowski problem for a broad range of p values.

Findings

Established long-time existence of the flow.

Proved convergence to solutions of the Minkowski problem.

Unified approach applicable to a wide class of curvature problems.

Abstract

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$