Parabolic stable surfaces with constant mean curvature

Abstract

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ 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 $H\in \mathbb{R}$ in homogeneous spaces $\mathbb{E}(\kappa ,\tau )$ with four dimensional isometry group. For instance, if M is an orientable, parabolic, complete immersed surface with constant mean curvature H in ${\mathbb{H}}^2\times \mathbb{R}$, then $|H|\le 1/2$ and if equality holds, then M is either an entire graph or a vertical horocylinder.

15 pages, 1 figure. Minor changes have been incorporated (exchange finite capacity by parabolicity, and simplify the proof of Theorem 1).

Cited by in corpus (5)

• Minimal surfaces with positive genus and finite total curvature in ${\mathbb{H}}^2\times \mathbb{R}$
• Compact stable surfaces with constant mean curvature in Killing submersions
• Height and area estimates for constant mean curvature graphs in E(κ,τ)-spaces
• Dual quadratic differentials and entire minimal graphs in Heisenberg space
• On complete constant mean curvature vertical multigraphs in E(κ,τ)