Parabolic stable surfaces with constant mean curvature

TL;DR

This paper investigates the properties of bounded solutions to certain Schrödinger operators on parabolic manifolds, deriving geometric consequences for stable constant mean curvature surfaces in specific homogeneous spaces.

Contribution

It establishes a uniqueness and zero-set property for solutions of Schrödinger operators on parabolic manifolds and applies these results to classify stable CMC surfaces in homogeneous spaces.

Findings

Bounded solutions are either identically zero or have no zeros.

Stable CMC surfaces in $ extbf{H}^2 imes extbf{R}$ with $|H|=1/2$ are classified as entire graphs or horocylinders.

The mean curvature $|H|$ is bounded above by 1/2 for such surfaces.

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 $|H|\leq 1/2$ and if equality holds, then M is either an entire graph or a vertical horocylinder.