A Colding-Minicozzi Stability inequality and its applications

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 non-negative potential on $\Sigma$, K the Gaussian curvature and $a$ is a non-negative 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 L is nonpositive acting on functions compactly supported on $\Sigma$ and we obtain results in the spirit of some theorems of Ficher-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.