Surfaces with maximal constant mean curvature

In this note we consider asymptotically flat manifolds withnon-negative scalar curvature and an inner boundary whichis an outermost minimal surface. We show that there existsan upper bound for the mean curvature of a constant meancurvature (CMC) surface homologous to a subset of theinterior boundary components. This bound allows us to finda maximizer for the CMC of a surface homologous to theinner boundary.

With this maximizer at hand, we can construct an increasingfamily of sets with boundaries of increasing CMC. Weinterpret this family as a weak version of a CMC foliation.

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Published 1 January 2010