Communications in Analysis and Geometry

Volume 30 (2022)

Number 7

An anisotropic shrinking flow and $L_p$ Minkowski problem

Pages: 1511 – 1540

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n7.a3

Authors

Weimin Sheng (School of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Caihong Yi (School of Mathematical Sciences, Zhejiang University, Hangzhou, China; and School of Mathematics, Hangzhou Normal University, Hangzhou, China)

Abstract

In this paper, we consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $R^{n+1}$ with speed $f u^\alpha \sigma^{-\beta}_n$, where $u$ is the support function of the hypersurface, $\alpha , \beta \in R^1$, and $\beta \gt 0, \sigma_n$ is the $n$-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow exists a unique smooth solution for all time and converges smoothly after normalisation to a smooth solution of the equation $ f u^\alpha \sigma^{-\beta}_n = c$ in the following cases $1- n \beta - 2 \beta \lt \alpha \lt 1 + n \beta$, $\alpha \neq 1 - \beta$ and $\alpha = 0, \beta = 1$ respectively, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on $S^n$. For the case $\alpha \geq 1 + n \beta , \beta \gt 0$, we prove that the flow converges smoothly after normalisation to a unique smooth solution of $f u^{\alpha-1} \sigma^{-\beta}_n = c$ without any constraint on the initial hypersuface and smooth positive function $f$. When $\beta = 1$, our argument provides a uniform proof to the existence of the solutions to the $L_p$ Minkowski problem $u^{1-p} \sigma_n = \phi$ for $p \in (-n - 1,+ \infty)$ where $\phi$ is a smooth positive function on $S^n$.

Received 2 July 2019

Accepted 2 March 2020

Published 25 May 2023