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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Colding-Minicozzi stability inequality and its applications


Authors: José M. Espinar and Harold Rosenberg
Journal: Trans. Amer. Math. Soc. 363 (2011), 2447-2465
MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
Published electronically: November 30, 2010
MathSciNet review: 2763722
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Abstract: We consider operators $ L$ acting on functions on a Riemannian surface, $ \Sigma$, of the form

$\displaystyle L=\Delta+V-a K .$

Here $ \Delta$ is the Laplacian of $ \Sigma$, $ V$ a nonnegative potential on $ \Sigma$, $ K$ the Gaussian curvature and $ a$ is a nonnegative constant.

Such operators $ L$ arise as the stability operator of $ \Sigma$ immersed in a Riemannian $ 3-$manifold with constant mean curvature (for particular choices of $ V$ and $ a$). We assume that $ L$ is nonpositive acting on functions compactly supported on $ \Sigma$ and we obtain results in the spirit of some theorems of Fischer-Colbrie-Schoen, Colding-Minicozzi and Castillon. We extend these theorems to $ a \leq 1/4$. We obtain results on the conformal type of $ \Sigma$ and a distance (to the boundary) lemma.


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Additional Information

José M. Espinar
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: jespinar@ugr.es

Harold Rosenberg
Affiliation: Instituto de Matematica Pura y Aplicada, 110 Estrada Dona Castorina, Rio de Janeiro 22460-320, Brazil
Email: rosen@impa.br

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05005-6
PII: S 0002-9947(2010)05005-6
Keywords: Stability, conformal type, Distance Lemma
Received by editor(s): December 10, 2008
Published electronically: November 30, 2010
Additional Notes: The author was partially supported by Spanish MEC-FEDER Grant MTM2007-65249, and Regional J. Andalucía Grants P06-FQM-01642 and FQM325.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.