Flow by Gauss curvature to the $L_p$-Gaussian Minkowski problem

TL;DR

This paper addresses the $L_p$-Gaussian Minkowski problem by establishing existence results using variational methods and curvature flows, advancing the understanding of convex hypersurface geometry in Gaussian spaces.

Contribution

It introduces new curvature flow techniques and variational approaches to solve the $L_p$-Gaussian Minkowski problem for various parameter ranges.

Findings

Proved existence of solutions for $p>0$ and $-n-1<p extless 0$ with even prescribed functions.

Constructed curvature flows that converge to solutions of the problem.

Provided a parabolic proof for cases $p extgreater n+1$ and $0 extless p extless n+1$.

Abstract

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the $L_p$-Gaussian Minkowski problem in cases of $p\geq n+1$ and $0<p<n+1$ with even prescribed function, respectively.