Finite index operators on surfaces

Original · EN

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 ∈ C∞(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 ∈ L¹(Σ) and P is non positive, or a decay condition with respect to a point p₀ ∈ Σ, i.e. |P(q)|≤ M/d(p₀,q) (where d is the distance function in Σ), we control the topology and conformal type of Σ. Moreover, we establish a Distance Lemma. We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion.