A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1

Abstract

Abstract In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ ( θ ) ϕ ⁢ ( ρ ) α − 2 ⁢ ϕ ⁢ ( ρ ) 2 + | ∇ ¯ ⁢ ρ | 2 , K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+\lvert\overline{\nabla}\rho\rvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 \alpha=n+1 ) in hyperbolic space, where f ~ \tilde{f} is a smooth positive function on S n \mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ⁢ ( ρ ) = sinh ⁡ ρ \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 α ≥ n + 1 \alpha\geq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 \mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2<\alpha\leq n+1 under the evenness assumption of f ~ \tilde{f} and prove the existence of solutions to the above equations.