A Colding-Minicozzi stability inequality and its applications

Espinar, José; Rosenberg, Harold

doi:10.1090/S0002-9947-2010-05005-6

A Colding-Minicozzi stability inequality and its applications
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by José M. Espinar and Harold Rosenberg PDF

Trans. Amer. Math. Soc. 363 (2011), 2447-2465 Request permission

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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