Parabolic stable surfaces with constant mean curvature

Abstract

Abstract. We prove that if u is a bounded smooth function in the kernel of a nonnega-tive Schrödinger operator −L = −( ∆ + q) on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of such functions is 1-dimensional. We obtain consequences for orientable, complete stable surfaces with constant mean curvature H ∈ R in homogeneous spaces E(κ, τ) with four dimensional isometry group. For instance, if M is an orientable, parabolic, complete immersed sur-face with constant mean curvature H in H2×R, then |H | ≤ 1 2 and if equality holds, then M is either an entire graph or a vertical horocylinder.