Stability and largeness properties of minimal surfaces in higher codimension

Abstract

We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in $\mathbb R^n$ lies in an even dimensional affine subspace and is holomorphic for some constant orthogonal complex structure. For stable minimal tori in Riemannian manifolds we give an explicit bound on the systole in terms of a positive lower bound on the isotropic curvature. As an application we estimate the systole of noncyclic abelian subgroups of the fundamental group of PIC manifolds. This gives a new proof of the result of [5] that the fundamental cannot contain a noncyclic free abelian subgroup. The proofs apply the structure theory of holomorphic vector bundles over genus 1 Riemann surfaces developed by M. Atiyah [2].