A result on large surfaces of prescribed mean curvature in a Riemannian manifold

Abstract

Plateau's problem (PP) is studied for surfaces of prescribed mean curvature spanned by a given contour in a 3-d Riemannian manifold. We consider the local situation where a neighborhood of a given point on the manifold is described by a single normal chart. Under certain conditions on H and the contour, existence of a small H -surface to (PP) is guaranteed by [HK]. The purpose of this paper is the investigation of large H -surfaces. Our result states: For sufficiently large (constant) mean curvature and a sufficiently small contour depending on the local geometry of the manifold, (PP) has at least two solutions, a small one and a large one. The proof is based on mountain pass arguments and uses -in contrast to results in the 3-d Euclidean space and in order to derive conformality directly -also a deformation constructed by variations of the independent variable.Mathematics Subject Classification: 53A10, 53A35