Abstract
We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures.
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Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces”.
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Izmestiev, I. A Variational Proof of Alexandrov’s Convex Cap Theorem. Discrete Comput Geom 40, 561–585 (2008). https://doi.org/10.1007/s00454-008-9077-7
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DOI: https://doi.org/10.1007/s00454-008-9077-7