Pure Maths Colloquium: Infinitesimal rigidity of surfaces and manifolds and the Hilbert-Einstein functional
7 February 2011 17:15 in CM221
By a classical theorem of Blaschke and Liebmann, every smooth strictly convex surface in R3 is infinitesimally rigid. The same is true for convex surfaces in the hyperbolic space.
On the other hand, results of Calabi, Weil, and Koiso imply infinitesimal rigidity of compact closed hyperbolic 3-manifolds.
One would like to unite the arguments of Koiso and Blaschke, which both use the Bochner technique, to reprove the infinitesimal rigidity of hyperbolic manifolds with convex boundary. (Already proved by Schlenker, but with a different method.) A possible advantage of a Bochner-type argument is the possibility to extend it to manifolds with non-smooth boundary. In particular, this could yield a proof of the Pleating Lamination Conjecture.
In this talk I will interpret Blaschke's proof in terms of variations of the Hilbert-Einstein functional, thus bringing it closer to Koiso's argument.
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