On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume.
Let \((M, \Phi)\) be a Finsler manifold acted upon by \(\mathbb{Z}^n\). For \(z\in \mathbb{Z}^n\) and \(x\in M\), the action of \(z\) on \(x\) is denoted by \(x+z\). \((M, \Phi)\) is said to be periodic if \(M/\mathbb{Z}^n\) is compact and the projection \(M\to M/\mathbb{Z}^n\) is a covering map. Assume that \((M, \Phi)\) is a periodic Finsler manifold. The stable norm of \((M, \Phi)\) is a norm \(\| \cdot \| \) on \(\mathbb{R}^n\) which is defined as follows. First, let \[ \| z\| :=\lim_{k\to \infty} \frac{d_{\Phi} (x, x+k z) } { k}, \qquad z\in \mathbb{Z}^n, \] where \( x\in M\) is a fixed point. The limit exists and does not depend on \(x\). Then it extends to a norm on \(\mathbb{R}^n\). The asymptotic volume growth \(\nu(M, \Phi)\) is defined by \[ \nu (M, \Phi) :=\lim_{r\to +\infty} \frac{\text{Vol} (B_r(x))}{r}, \] where \(B_r(x)\) is the metric ball of a radius \(r\) around \(x\) and Vol is the Hausdorff measure or the symplectic volume. It is proved by D. Burago that for a periodic Finsler manifold \((M, \Phi)\), there is a constant \(C\) such that \[ \| z\| \leq d_{\Phi}(x, x+ z ) \leq \| z\| + C, \qquad z\in \mathbb{Z}^n, \;x\in M. \] There are three conjectures on the stable norm and asymptotic volume growth of periodic Finsler manifolds. Conjecture A: Let \((M,\Phi)\) be the universal cover of a Finsler \(n\)-torus (i.e., \(\Phi\) is a Finsler metric on \(M=\mathbb{R}^n\) invariant under \(\mathbb{Z}^n\) acting by translations). Then \(\nu(M, \Phi) \geq \nu ( \mathbb{R}^n\), \(\| \cdot \| )\). Conjecture B: Let \(X\) be a finite-dimensional Banach space, \(V \subset X\) an \(n\)-dimensional affine subspace, \(A\subset V\) a region diffeomorphic to the disc \(D^n\), and \(f: D^n \to X\) an immersion such that \(f| _{\partial D^n}\) is a diffeomorphism onto \(\partial A\). Then \(\text{Vol}(f) \geq \text{Vol}(A)\). Conjecture C: Let \(A\) be a bounded region in an \(n\)-dimensional Banach space \((V^n, \| \cdot \| )\). Let \(\Phi\) be a Finsler metric on \(A\) such that \[ d_{\Phi}(x,y) \geq \| x- y\| , \qquad x,y\in \partial A. \] Then \(\text{Vol}(A, \Phi) \geq \text{Vol} (A, \| \cdot \| )\). The main result in this paper is that the three conjectures A, B and C are equivalent in any dimension \(n\).
scientific article; zbMATH DE number 1877085
scientific article