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.
Categories: math.DG math.AP