Research Summary

In my doctoral dissertation (directed by W. P. Thurston) I studied the geometry of convex polyhedra in hyperbolic 3-space H3, and succeeded in producing a geometric characterization of dihedral angles of compact convex polyhedra by reducing the question to a convex isometric embedding problem in the De Sitter sphere, and resolving this problem. In particular, this produced a simple alternative proof of Andre’ev’s celebrated characterization of acute-angled hyperbolic polyhedra ([43, 31]). I later developed the ideas of [31] to characterize the angles of polyhedra with some (or all) ideal vertices in H3 ([33, 24]). It turned out that an easy corollary of this result resolved a problem on characterizing the combinatorial types of convex polyhedra inscribed in the sphere, posed by Jakob Steiner in 1832. Itwas also central to combinatorial and computational geometry, since there is a natural correspondence between ideal polyhedra and Delaunay tessellations of pointsets in the Euclidean plane. The methods of [24] were completely geometric, use a fairly delicate limiting argument, and failed to produce a uniqueness result. Search for this uniqueness result resulted in my paper [30], which took a completely analytic approach (by studying affine structures and a hyperbolic volume functional), and succeeded in producing a sought-after uniqueness proof, and improved understanding of the geometry and combinatorics of ideal polyhedra and singular Euclidean surfaces. A particular corollary of those results was the observation that the hyperbolic volume is a convex function on polyhedra of a fixed combinatorial type, and that the maxima of this function often correspond to arithmetic surfaces. Some of these ideas were further developed in [17]. Some of the results of the latter paper are the following: