Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces

Abstract

It is proved that if M is a rotationally symmetric Hadamard surface which is conformally equivalent to the hyperbolic disk then the asymptotic Dirichlet problem for the minimal surface equation is uniquely solvable for any continuous asymptotic boundary data. This result gives a partial answer of a question in Galvez and Rosenberg (Am J Math 132:1249–1273, 2010) about the existence of entire minimal graphs on Hadamard surfaces with sectional curvature possibly degenerating at infinity.