Finite index operators on surfaces

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

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