Gaussian Images of Surfaces and Ellipticity of Surface Area Functionals

Before we proceed with rigorous formulations and applications, we want to give a very informal and “easy-to-visualize” description of one of the central results of the paper. Assume that we are given an n-plane P ⊂ RN , n < N , a collection of oriented n-planes ei, i = 1, 2, . . . k, and a collection of positive reals ai, i = 1, 2, . . . k. We want to know whether there exists an oriented polyhedral surface S such that its boundary belongs to P , each of its faces is parallel to one of ei’s (and has the same orientation), and the total area of all faces parallel to ei is ai. If N = n + 1, the answer is given by the classic Existence Theorem of Minkowski. Let ni be the unit normal to ei (where the orientation of ei determines one of the two choices for a unit normal). Then such a surface S exists if and only if the following obviously necessary “linear algebra” condition is satisfied: ei’s do not lie in a hyperplane and their weighted sum ∑ aini is normal to P . Furthermore, then S can be chosen among convex surfaces. In general case, already if n = 2 and N = 4, no condition like the Minkowski “linear algebra” condition suffices: in generic position ei’s may have no lines in common, so one cannot construct any polyhedral surface of faces parallel to ei’s. Hence we reformulate the problem as follows: given P , ei’s and ai’s, we wonder if, for every positive ε, there is a surface such that the total area of its faces parallel to each ei is ai, and the total area of all other faces is less than ε. Now the answer to this question depends on what we exactly mean by a polyhedral surface. If one considers immersed PL-manifolds, we show that, like in the Existence Theorem of Minkowski, such surfaces can always be found provided the obvious “linear algebra” condition similar to that in the theorem of Minkowski is satisfied. However, if one insists on an embedded surface, or at least on an immersed surface whose boundary is embedded in P , there are other (not really well understood) constraints.