A Variational Proof of Alexandrov’s Convex Cap Theorem
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- Izmestiev, I. Discrete Comput Geom (2008) 40: 561. doi:10.1007/s00454-008-9077-7
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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.