A flow approach to the prescribed Gaussian curvature problem in $\mathbb{H}^{n+1}$

TL;DR

This paper introduces a flow method to solve a generalized prescribed Gaussian curvature problem in hyperbolic space, establishing existence and uniqueness results for certain parameter ranges and providing a new proof for the classical Alexandrov problem.

Contribution

The paper develops a flow-based approach to solve the prescribed Gaussian curvature problem in hyperbolic space, extending results to a broader class of equations and offering a parabolic proof for the Alexandrov problem.

Findings

Existence and uniqueness of solutions for lpha  when lpha .

Solutions exist for 2 < lpha  under evenness assumption of .

Provides a parabolic proof for the classical Alexandrov problem in hyperbolic space.

Abstract

In this paper, we study the following prescribed Gaussian curvature problem $$K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^2+|\bar{\nabla}\rho|^2}},$$ a generalization of the Alexandrov problem ($\alpha=n+1$) in hyperbolic space, where $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$, $\rho$ is the radial function of the hypersurface, $\phi(\rho)=\sinh\rho$ and $K$ is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when $\alpha\geq n+1$. Our argument provides a parabolic proof in smooth category for the Alexandrov problem in $\mathbb{H}^{n+1}$. We also consider the cases $2<\alpha\leq n+1$ under the evenness assumption of $\tilde{f}$ and prove the existence of solutions to the above equations.