# 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.

## Key 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.