Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

Variational properties of the discrete Hilbert-Einstein functional

Ivan Izmestiev

Slides can be found by following the link.

For a 3-manifold triangulated by Euclidean simplices, the discrete Hilbert-Einstein functional HE is defined as the sum of edge lengths times the curvatures around the edges. It became known as Regge's action in quantum gravity, but the main ideas can be traced back to Steiner and Schlaefli.

The main feature of HE is that the partial derivative with respect to an edge length equals the curvature of that edge. This makes HE suitable to study rigidity and geometrization questions: there is an analog of HE for triangulations by hyperbolic simplices, and the variational property implies that critical points of HE are space-forms.

We will give an overview of old and new results on the signature of the Hessian of HE with applications to rigidity problems.

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