A Colding-Minicozzi stability inequality and its applications
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- by José M. Espinar and Harold Rosenberg PDF
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Abstract:
We consider operators $L$ acting on functions on a Riemannian surface, $\Sigma$, of the form \[ L=\Delta +V-a K .\]
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
- MR Author ID: 150570
- Email: rosen@impa.br
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 2447-2465
- MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
- DOI: https://doi.org/10.1090/S0002-9947-2010-05005-6
- MathSciNet review: 2763722