TITLE

# Parabolic stable surfaces with constant mean curvature

AUTHOR(S)
Manzano, José; Pérez, Joaquín; Rodríguez, M.
PUB. DATE
September 2011
SOURCE
Calculus of Variations & Partial Differential Equations;Sep2011, Vol. 42 Issue 1/2, p137
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We prove that if u is a bounded smooth function in the kernel of a nonnegative SchrÃ¶dinger operator âˆ’ L = âˆ’(Î” + 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 $${|H|\leq \frac{1}{2}}$$ and if equality holds, then M is either an entire graph or a vertical horocylinder.
ACCESSION #
62869042

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