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

Abstract

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

arXiv comment31 pages, comments welcome