Discrete & Computational Geometry

, Volume 40, Issue 4, pp 561–585

A Variational Proof of Alexandrov’s Convex Cap Theorem


    • Institut für MathematikTechnische Universität Berlin

DOI: 10.1007/s00454-008-9077-7

Cite this article as:
Izmestiev, I. Discrete Comput Geom (2008) 40: 561. doi:10.1007/s00454-008-9077-7


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.


Convex capDiscrete Hilbert–Einstein functionalEuclidean cone metric

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© Springer Science+Business Media, LLC 2008