A unified flow approach to smooth $L^p$ Christoffel-Minkowski problem for $p>1$

TL;DR

This paper introduces a unified flow method to solve the smooth $L^p$ Christoffel-Minkowski problem for $p>1$, demonstrating long-term existence and convergence to solutions under certain conditions.

Contribution

It develops a new flow approach that unifies the treatment of the $L^p$ Christoffel-Minkowski problem for $p>1$, proving convergence to solutions.

Findings

Flow exists for all time under certain conditions.

Flow converges smoothly to the solution of the $L^p$ Christoffel-Minkowski problem.

Provides a new method for solving the problem for $p>1$.

Abstract

In this paper we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed $\psi\sigma_k(\lambda)^{\alpha}$, where $\alpha$ is a positive constant, $\sigma_k(\lambda)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $\psi$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $\psi$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}( x ) \sigma_k \left( \nabla^2u+uI\right)=c\psi(x)$ for $p>1$.