On Asymptotic Volume of Finsler Tori, Minimal Surfaces in Normed Spaces, and Symplectic Filling Volume

Abstract

The main unconditional result of this paper, Theorem 3, states that every two-dimensional affine disc in a normed space (that is, a disc contained in a two-dimensional affine subspace) is an area-minimizing surface among all immersed discs with the same boundary, with respect to the symplectic (Holmes-Thompson) surface area. To emphasize that this is not at all obvious, it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect (Theorem 2), and that it is not known for surfaces that may not be topological discs. The result still may not sound too exciting to the reader who never looked at the problem before, even though it goes back to Busemann's works in the 50's (see [BES], [Th] and references there), and the proof heavily relies on asymptotic geometry of tori. We believe that it is more important that we embed this problem into a whole area of (mostly open) problems, as well as give some partial results and suggest certain directions of how to attack them. We begin with a trivial statement: