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In this paper, we study the existence and uniqueness of solutions to Jenkins–Serrin type problems on domains in a Riemannian surface: infinite boundary data are allowed. In the case of unbounded domains, the study is focused on the hyperbolic plane. A counterexample to uniqueness is given for entire minimal graphs over ℍ 2 .

The Dirichlet problem for the minimal surface equation, with possible infinite boundary data, over domains in a Riemannian surface