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In this note, we study the asymptotic Plateau problem in hyperbolic space, and we prove the existence of a smooth complete hypersurface of constant scalar curvature in hyperbolic space with a prescribed asymptotic boundary at infinity. Following a pioneering work of Bo Guan and Joel Spruck, we seek the solution as a vertical graph over a bounded domain and solve the corresponding Dirichlet problem for a fully nonlinear partial differential equation by establishing the crucial second order estimates for admissible solutions. Our proof consists of three main ingredients: (1) a non-standard, special choice for the parameter in the auxiliary function, (2) a so-called almost-Jacobi inequality for the equation operator, and (3) a set of arguments which reduce the situation to semi-convex case and which keep the coefficient of the troublesome negative term within a suitable magnitude.

Author

Wang, Bin

Published

2024

Language

EN