Finite Index Operators on Surfaces

Abstract

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the formwhere ∆ 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 non positive, 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 ⊂ H 2 × R, H > 1/2, so that either K + e := max {0, K e } ∈ L 1 (Σ) 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 Σ ⊂ E(κ, τ ), 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:Berger , there are no such a stable H−surface.-In Nil 3 , H = 0 and Σ is either a vertical plane (i.e. a vertical cylinder over a straight line in R 2 ) or an entire vertical graph.-In PSL(2, R), H = √ −κ/2 and Σ is either a vertical horocylinder (i.e. a vertical cylinder over a horocycle in H 2 (κ)) or an entire graph.

Height and Area Estimates for Constant Mean Curvature Graphs in $$\mathbb {E}(\kappa ,\tau )$$ E ( κ , τ ) -Spaces

“…It is worth emphasizing that our gradient estimate for Σ is sharper than the general estimates for the angle function in Killing submersions given by Rosenberg, Souam and Toubiana in [29]. As a consequence, we improve a result of Espinar [13,Corollary 5.2] by showing that a complete orientable stable surface with constant mean curvature H in E(κ, τ ), with τ = 0 and 4H 2 + κ ≥ 0, whose angle function is square-integrable must be a vertical cylinder (Corollary 4).…”

“…Proof. Under these hypothesis, [13,Corollary 5.1] implies that κ ≤ 0 and Σ has critical mean curvature, and it is either a vertical cylinder or an entire graph. By the Daniel correspondence, we will suppose that E(κ, τ ) = Nil 3 (τ ) without loss of generality (stability is also preserved by the correspondence, see [8,Proposition 5.12]).…”

Gradient Schrödinger operators, manifolds with density and applications

“…Most of the above results can be generalized to the case that Σ has finite index, that is, there is only a finite dimensional space of normal variations which strictly decrease the weighted area (see [2]). Moreover, we could relax the hypothesis on the P-scalar curvature by either an integrability condition or a decay condition as in [4] or [16].…”

“…The above Main Lemma allows us to use the strong machinery developed for Schrödinger operators since the pioneer work of T. Colding and W. Minicozzi [11] and improved in [6,15,16,17,34] and references therein.…”

Inverse spectral positivity for surfaces

“…c = inf q > 0. Theorem 1.2 can be extended to this situation, improving the results of [7]. We refer to [1] for the details.…”