Uniqueness of solutions to some classes of anisotropic and isotropic curvature problems

TL;DR

This paper proves the uniqueness of solutions for various anisotropic and isotropic curvature problems using integral formulas and geometric analysis, including applications to Minkowski and Gaussian-Minkowski problems.

Contribution

It introduces new uniqueness results for a range of curvature problems, extending existing methods and applying them to generalized Minkowski-type equations.

Findings

Uniqueness of smooth admissible solutions to Orlicz-Minkowski problems.

Solutions to certain isotropic curvature equations are proven to be origin-centered spheres.

Establishment of uniqueness for isotropic Gaussian-Minkowski and dual Minkowski problems.

Abstract

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern \cite{Ch59}, we obtain the uniqueness of smooth admissible solutions to a class of Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon's uniqueness result \cite{Si67}, we then prove that the only smooth strictly convex solution to the following isotropic curvature problem \begin{equation}\label{ab-1} \left(\frac{P_k(W)}{P_l(W)}\right)^{\frac{1}{k-l}}=\psi(u,r)\quad \text{on}\ \mathbb{S}^n \end{equation} must be an origin-centred sphere, where $W=(\nabla^2 u+u g_0)$, $\partial_1\psi\ge 0,\partial_2\psi\ge 0$ and at least one of these inequalities is strict. As an application, we establish the uniqueness of solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the uniqueness result for the following isotropic $L_p$ dual Minkowski problem \begin{equation}\label{ab-2} u^{1-p} r^{q-n-1}\det(W)=1\quad \text{on}\ \mathbb{S}^n, \end{equation} where $-n-1<p\le -1$ and $n+1\le q\le n+\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{(1+p)(n+1+p)}{n(n+2)}}$. This result utilizes the method developed by Ivaki and Milman \cite{IM23} and generalizes a result due to Brendle, Choi and Daskalopoulos \cite{BCD17}.