Polyhedral Surfaces and Combinatorial Curvature
A polyhedral surface R is obtained from a (possibly infinite) 2-cell embedded graph G by realizing each face f by a ``flat'' polygon Pf in the Euclidean plane.
If two faces f,g share an edge e, then the sides of Pf and Pg corresponding to e are required to have the same length. The corresponding polyhedral surface is then obtained from the collection of all such polygons by identifying them along sides corresponding to the edges of the graph. Such polyhedral surface is endowed with a metric which is locally Euclidean, with the exception of the vertices.
Let V be the set of vertices of the embedded graph G, and for every v € V, let k(v) denote the (Gaussian) curvature of v: 2 ∏ minus the sum of incident polygon angles. Descartes showed that ∑v €Vk(v) = 4 ∏ whenever R may be realized as the surface of a convex polytope in the 3-space. This equality can be extended to arbitrary surfaces (Gauss-Bonnet) and even to infinite embedded graphs.
Several results will be discussed, including Alexandrov's Theorem, Higuchi Conjecture and its solution, representations of fullerenes, and eigenvalues of (3,6)-polyhedra.
Bojan Mohar, Simon Fraser University, Canada and University of Ljubljana, Slovenia