Parametric surfaces of least \(H\)-energy in a Riemannian manifold
The main aim of this paper is to establish existence theorems for the Plateau problem with prescribed mean curvature in Riemannian \(3\)-manifolds, going beyond the results of \textit{R. Gulliver} [J. Differ. Geom. 8, 317-330 (1973; Zbl 0275.53033)] and \textit{St. Hildebrandt} and \textit{H. Kaul} [Commun. Pure Appl. Math. 25, 187-223 (1972; Zbl 0245.53006)], more precisely, by renouncing on some of the conditions that the cited authors put on the boundary. Hence, in the present paper, the considered surfaces are not contained in a given coordinate domain of the manifold and the existence results are proven for all curves bounding a surface of sufficiently small area. The authors first prove a general existence theorem for weak solutions of the Plateau problem. Then, they combine it with conditions in Riemannian manifolds and some results on isoperimetric inequalities [same authors, Calc. Var. Partial Differ. Equ. 1, No. 4, 355-406 (1993; Zbl 0806.49028)] to obtain geometric conditions that the Jordan boundary curve and the mean curvature function must satisfy in order to assure the existence of the solution. The paper ends with some brief comments on proving the regularity of the solutions obtained before.
scientific article; zbMATH DE number 1337013