A blow-up method to prescribed mean curvature graphs with fixed boundaries

Abstract

In this paper we discuss the existence of prescribed mean curvature (PMC) graphs with fixed graphical boundaries in the product manifold N×R. We define an Nc-f domain in which closure does not contain certain domains with the mean curvature of its boundary equal to f . We show the existence of corresponding PMC graphs over bounded Nc-f domains with natural boundary conditions via a blow-up method from Schoen-Yau. This existence can be used in the Plateau problem of PMC surfaces in 3-manifolds.