Optimal regularity of constant curvature graphs in Hyperbolic space

Abstract

Inspired by Han and Jiang (Boundary expansions for minimal graphs in the hyperbolic space, 2014. arXiv:1412.7608 , The convergence of boundary expansions and the analyticity of minimal surfaces in the hyperbolic space, 2018. arXiv:1801.08348 ) we study the boundary regularity of constant curvature hypersurfaces in the hyperbolic space $${\mathbb {H}}^{n+1}$$ , which have prescribed asymptotic boundary at infinity. Through constructing the boundary expansions of the solutions, we derive the optimal regularity of the solutions. Moreover, we obtain an equivalent condition that guarantees the smoothness of the solutions.