  We consider differential operators $L$ acting on functions on a Riemannian
surface, $\Sigma$, of the form $$L = \Delta + V -a K ,$$where $\Delta$ is the
Laplacian of $\Sigma$, $K$ is the Gaussian curvature, $a$ is a positive
constant and $V \in C^{\infty}(\Sigma)$. Such operators $L$ arise as the
stability operator of $\Sigma$ immersed in a Riemannian three-manifold with
constant mean curvature (for particular choices of $V$ and $a$).
  We assume $L$ is nonpositive acting on functions compactly supported on
$\Sigma$. If the potential, $V:= c + P $ with $c$ a nonnegative constant,
verifies either an integrability condition, i.e. $P \in L^1(\Sigma)$ and $P$ is
non positive, or a decay condition with respect to a point $p_0 \in \Sigma$,
i.e. $|P(q)|\leq M/d(p_0,q)$ (where $d$ is the distance function in $\Sigma$),
we control the topology and conformal type of $\Sigma$. Moreover, we establish
a {\it Distance Lemma}.
  We apply such results to complete oriented stable $H-$surfaces immersed in a
Killing submersion.
