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A variational proof of Alexandrov's convex cap theorem

Ivan Izmestiev
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Abstract

We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures.

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Ivan Izmestiev. 2007-04-24. A variational proof of Alexandrov's convex cap theorem. https://arxiv.org/abs/math/0703169

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