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