Abstract
We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrödinger operator −L = −(Δ + 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.
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Manzano, J.M., Pérez, J. & Rodríguez, M.M. Parabolic stable surfaces with constant mean curvature. Calc. Var. 42, 137–152 (2011). https://doi.org/10.1007/s00526-010-0383-6
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DOI: https://doi.org/10.1007/s00526-010-0383-6