Closed Hypersurfaces of Prescribed Mean Curvature in Locally Conformally Flat Riemannian Manifolds

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to S for small n, n ≤ 6, provided there are barriers. 0. Introduction In a complete (n+1)-dimensional manifold N we want to find closed hypersurfaces M of prescribed mean curvature. To be more precise, let Ω be a connected open subset of N , f ∈ C(Ω̄), then we look for a closed hypersurface M ⊂ Ω such that (0.1) H|M = f(x) ∀x ∈ M, where H|M is the mean curvature, i.e. the sum of the principal curvatures. The existence of a generalized solution M = ∂E, where E is a Caccioppoli set minimizing an appropriate functional is easily demonstrated if the boundary of Ω is supposed to consist of two components acting as barriers. For small n, n ≤ 6, the generalized solution is also a classical one, since it is smooth, M ∈ C, and hence a solution of (0.1); but nothing is known about its topological type. We shall prove that in the case when n ≤ 6 and N is locally conformally flat, or more precisely, when in Ω the metric is conformally flat, smooth solutions homeomorphic to S exist. We make the following definition Definition 0.1. Let M1, M2 be closed hypersurfaces in N homeomorphic to S n and of class C which bound an open, connected, relatively compact subset Ω. M1, M2 are called barriers for (H, f) if (0.2) H|M1 ≤ f 1991 Mathematics Subject Classification. 35.