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

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 the problem goes back to Busemann’s works in the 50th (see [BES], [Th] and references there), and the proof heavily relies on asymptotic geometry of tori. We belive 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 attack them. We begin with a trivial statement: Minimality of Flats in Euclidean Spaces. A ball in an n-dimensional affine subspace of a Euclidean space has the smallest n-dimensional surface area among all n-dimensional immersed discs with the same boundary. The proof boils down to considering the orthogonal projection onto the affine subspace, which is an area non-increasing map. Of course, the statement remains true (and obvious) if “immersed discs” are substituted by “immersed surfaces” or “Lipschits singular chains with real coefficients”. The next statement is an easy corollary of Besicovitch’s inequality: Riemannian Filling Volume in Euclidean Spaces. The Euclidean volume of a bounded region in a Euclidean space is less than or equal to the volume of this region with respect to any Riemannian metric whose distances between boundary points majorize those of the Euclidean distance function. Now let us formulate a volume growth theorem which is the main result of [BI1] and served as a starting point for this research.