Uniqueness of solutions to some classes of anisotropic and isotropic curvature 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}.