Hypersurfaces With Constant Mean Curvature And Prescribed Area

Mathematics Research Report MRR95-075

Hypersurfaces With Constant Mean Curvature And Prescribed Area

Frank Duzaar

Abstract: We study, in the setting of geometric measure theory, hypersurfaces T (of codimension one) with prescribed boundary B in Euclidean n+1 space which maximize volume (i.e. T together with a fixed hypersurface T encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity to solutions T of such variational problems and we show that, on the regular part of its support, T is a classical hypersurface of constant mean curvature. We also prove that T becomes more and more spherical as the value m of the mass constraint approaches $\infty$.

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