Stable minimal surfaces in M × ℝ

Abstract

In this paper, we classify the stable properly embedded orientable minimal surfaces in M × ℝ, where M is a closed orientable Riemannian surface. We show that such a surface is a product of a stable embedded geodesic on M with ℝ, a minimal graph over a region of M bounded by stable geodesics, M × {t} for some t ∈ ℝ, or is in a moduli space of periodic multigraphs parametrized by P × ℝ⁺, where P is the set of primitive (non-multiple) homology classes in H₁(M).