Discrete & Computational Geometry

, Volume 40, Issue 4, pp 561–585

A Variational Proof of Alexandrov’s Convex Cap Theorem


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 cap Discrete Hilbert–Einstein functional Euclidean cone metric 

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations