Mathematics Colloquium, Fall 2007

Two connections between combinatorial and differential geometry

There is a rich interplay between combinatorial and differential geometry. We will give first a geometric proof of a combinatorial result, and then a combinatorial analysis of a geometric moduli space. The first is joint work with Ivan Izmestiev, Rob Kusner, Guenter Rote, and Boris Springborn; the second with Karsten Grosse-Brauckmann, Nick Korevaar and Rob Kusner.

In any triangulation of the torus, the average vertex valence is 6. Can there be a triangulation where all vertices are regular (of valence 6) except for one of valence 5 and one of valence 7? The answer is no. We prove this geometrically, giving the torus the metric where each triangle is equilateral and then explicitly analyzing the holonomy. Indeed, techniques from Riemann surfaces can characterize exactly which euclidean cone metrics have full holonomy group no bigger than their restricted holonomy group (at least when the latter is finite).

Next we consider the moduli space M_k of Alexandrov-embedded surfaces of constant mean curvature which have k ends and genus 0 and are contained in a slab. We showed earlier that M_k is homeomorphic to an open manifold D_k of dimension 2k-3, defined as the moduli space of spherical metrics on an open disk with exactly k completion points. In fact, D_k is the ball B^{2k-3}; to show this we use the Voronoi diagram or Delaunay triangulation of the k completion points to get a tree, labeled by logarithms of cross-ratios. The combinatorics of the tree are tracked by the associahedron, and the labels give us a complexification of the cone over its dual. We note similarities to the spaces of labeled trees used in phylogenetic analysis.