  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.
