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.