Finite Index Operators on Surfaces

Abstract

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the form $$L = \Delta+ V -a K,$$ 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∈L 1(Σ) and P is nonpositive, or a decay condition with respect to a point p 0∈Σ, i.e., |P(q)|≤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 Distance Lemma.We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion. In particular, for stable H-surfaces in a simply-connected homogeneous space with 4-dimensional isometry group, we obtain: There are no complete stable H-surfaces Σ⊂ℍ2×ℝ, H>1/2, so that either \(K_{e}^{+}:=\max \left \{0,K_{e}\right \} \in L^{1} (\Sigma)\) or there exist a point p 0∈Σ and a constant M so that |K e (q)|≤M/d(p 0,q); here K e denotes the extrinsic curvature of Σ. Let \(\Sigma\subset \mathbb{E}(\kappa, \tau)\), τ≠0, be an oriented complete stable H-surface so that either ν 2∈L 1(Σ) and 4H 2+κ≥0, or there exist a point p 0∈Σ and a constant M so that |ν(q)|2≤M/d(p 0,q) and 4H 2+κ>0. Then: In \(\mathbb{S}^{3}_{\text{Berger}}\), there are no such a stable H-surfaces. In Nil3, H=0 and Σ is either a vertical plane (i.e., a vertical cylinder over a straight line in ℝ2) or an entire vertical graph. In \(\widetilde{\mathrm{PSL}(2,\mathbb{R})}\), \(H=\sqrt{-\kappa }/2\) and Σ is either a vertical horocylinder (i.e., a vertical cylinder over a horocycle in ℍ2(κ)) or an entire graph.