Discrete & Computational Geometry

, Volume 40, Issue 4, pp 561–585

A Variational Proof of Alexandrov’s Convex Cap Theorem

Authors

    • Institut für MathematikTechnische Universität Berlin
Article

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

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.

Keywords

Convex capDiscrete Hilbert–Einstein functionalEuclidean cone metric

Copyright information

© Springer Science+Business Media, LLC 2008