Geometry and Topology Seminar

Abstract: We will discuss the following problem: which measures can arise as the images of the surface area of closed surfaces (or of surfaces with planar boundaries) under the Gauss map? For instance, here is a very down-to-Earth formulation of basically the same problem: under what conditions can one construct a polyhedron (for instance, a 2-dimensional polyhedral surface in $\R^4$) with given areas and directions of its faces and no boundary (or, say, with a boundary lying in a given two-dimensional plane - a ``roof'' over that boundary curve)? We will give an answer to this problem, which depends on what we exactly mean by ``surfaces'', ``polyhedral surfaces'', and what types of boundaries are allowed. We will then discuss how this problem is related to ellipticity properties of surface area functionals, and certain Besicovich-type inequalities. If time permits, I will explain how this topic grew from the proof of the Hopf conjecture via looking at minimal surfaces in normed spaces. The talk is based on a joint work with Sergei Ivanov.