Rigidity of stable cylinders in three-manifolds

In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manifold has infinite volume.


Introduction
A stable compact domain Σ on a minimal surface in a Riemannian three-manifold M, is one whose area can not be decreased up to second order by a variation of the domain leaving the boundary fixed. Stable oriented domains Σ are characterized by the stability inequality for normal variations ψN [11] for all compactly supported functions ψ ∈ H 1,2 0 (Σ). Here |A| 2 denotes the the square of the length of the second fundamental form of Σ, Ric M (N, N) is the Ricci curvature of M in the direction of the normal N to Σ and ∇ is the gradient w.r.t. the induced metric.
One writes the stability inequality in the form where L is the linearized operator of the mean curvature L = ∆ + |A| 2 + Ric M .
In terms of L, stability means that −L is nonnegative, i.e., all its eigenvalues are nonnegative. Σ is said to have finite index if −L has only finitely many negative eigenvalues.
From the Gauss Equation, one can write the stability operator as L = ∆ − K + V , where ∆ and K are the Laplacian and Gauss curvature associated to the metric g respectively, and V := 1/2|A| 2 + S, where S denotes the scalar curvature associated to the metric g.
The index form of these kind of operators is where ∇ and · are the gradient and norm associated to the metric g. Thus, if Σ is stable, we have or equivalently In a seminar paper [5], D. Fischer-Colbrie and R. Schoen proved: Theorem A: Let M be a complete oriented three-manifold of non-negative scalar curvature. Let Σ be an oriented complete stable minimal surface in M. If Σ is noncompact, conformally equivalent to the cylinder and the absolute total curvature of Σ is finite, then Σ is flat and totally geodesic.
And they state [5,Remark 2]: We feel that the assumption of finite total curvature should not be essential in proving that the cylinder is flat and totally geodesic.
Recently, this question was partially answered in [3] under the assumption that the positive part of the Gaussian curvature is integrable, i.e. K + := max {0, K} ∈ L 1 (Σ), and totally answered by M. Reiris [10], he proved: Theorem B: Let M be a complete oriented three-manifold of non-negative scalar curvature. Let Σ be an oriented complete stable minimal surface in M diffeomorphic to the cylinder, then Σ is flat and totally geodesic.
Besides, Bray, Brendle and Neves [1] were able of determining the structure of a threemanifold M under the assumption of the existence of an area minimizing two-sphere. Specifically, they proved:

Then,
where R denotes the scalar curvature of M. Moreover, if the equality holds, then the universal cover of M is isometric to the standard cylinder S 2 × R up to scaling.
In this paper, we will go further. We will see how the existence of a stable cylinder verifying a bifurcation phenomena determines the ambient manifold M. First, let us make clear what we mean by bifurcation phenomena: • For each t ∈ (−δ, δ), the surface We should point out the condition that Σ bifurcates is necessary. In fact, one can construct the following example: Let C(−l, l) be the right cylinder of height 2l and radius 1 endowed with the flat metric. Close it up with two spherical caps S i , i = 1, 2 (one on the top and another on the bottom). Now, smooth the surface M 2 = C(−l, l) ∪ S 1 ∪ S 2 so that it is flat on C(−l + ε, l − ε), for some ε > 0, and has nonnegative Gaussian curvature.
Consider the three-manifold M 3 = M 2 × R. One can see that, if we take a closed geodesic γ(t) ⊂ C(−l + ε, l − ε) , t ∈ (−l + ε, l − ε), the surface Σ(t) := γ(t) × R is a complete stable minimal cylinder in M that bifurcates, but, when we reach t = l − ε, this property it might disappear (it could bifurcate as constant mean curvature surfaces at one side, but not minimal).
One interesting consequence of Theorem 1.1 is the following: Actually, the above conclusion (that is, the above Corollary 1.1) is also valid when the cylinder bifurcates only at one side.

Preliminaries
We denote by M a complete connected orientable Riemannian three-manifold, with Riemannian metric g. Moreover, throughout this work, we will assume that its scalar curvature is nonnegative, i.e., S ≥ 0. Σ ⊂ M will be assumed to be connected and oriented.
We denote by N the unit normal vector field along Σ. Let p 0 ∈ Σ be a point of the surface and D(p 0 , s), for s > 0, denote the geodesic disk centered at p 0 of radius s. We assume that D(p 0 , s) ∩ ∂Σ = ∅. Moreover, let r be the radial distance of a point p in D(p 0 , s) to p 0 . We write D(s) = D(p 0 , s).
We also denote l(s) = Length(∂D(s)) a(s) = Area(D(s)) Let Σ ⊂ M be a stable minimal surface diffeomorphic to the cylinder, then, from Theorem B [10], Σ is flat and totally geodesic. We will give a (more general) proof of this result in the abstract setting of Schrödinger-type operators: Lemma 2.1. Let Σ be a complete Riemannian surface. Let L = ∆ + V − aK be a differential operator on Σ acting on compactly supported f ∈ H 1,2 0 (Σ), where a > 1/4 is constant, V ≥ 0, ∆ and K are the Laplacian and Gauss curvature associated to the metric g respectively.
Assume that Σ is homeomorphic to the cylinder and −L is non-negative. Then, V ≡ 0 and K ≡ 0, therefore, i.e., its kernel is the constant functions. Here, L denotes the Jacobi operator.
Proof. Set b ≥ 1 and let us consider the radial function where r denotes the radial distance from a point p 0 ∈ Σ. Then, from [3, Lemma 3.1] (see also [9]), we have • Step 1: V vanishes identically on Σ.
Suppose there exists a point p 0 ∈ Σ so that V (p 0 ) > 0. From now on, we fix the point p 0 . Then, there exists ǫ > 0 so that V (q) ≥ δ for all q ∈ D(ǫ) = D(p 0 , ǫ). Since Σ is topollogically a cylinder, there exists s 0 > 0 so that for all s > s 0 we have χ(s) ≤ 0 (see [2,Lemma 1.4]). Now, from the above considerations, there exists β > 0 so that But, following [3], we can see that which is a contradiction. Thus, V vanishes identically along Σ.
On the one hand, the respective Gaussian curvatures are related by On the other hand, since Σ is topologically a cylinder, the Cohn-Vossen inequality says that is,K vanishes identically.
Thus, K = α∆ ln u. From this last equation, we get: that is, This last equation implies that u is constant, and since u satisfies Lu = 0, we have that K vanishes identically on Σ. In particular, Σ is parabolic (see [6,Lemma 5]) This implies that the Jacobi operator becomes L := ∆, and so the constant functions are in the kernel. But, since Σ is parabolic, such a kernel has dimension one (see [8]), therefore Set C := S 1 × R the flat cylinder, then we can parametrize Σ as the isometric immersion ψ 0 : C → M where Σ := ψ 0 (C). Also, set N 0 : C → NΣ the unit normal vector field along Σ.
For each t ∈ (−δ, δ), the lapse function ρ t : Σ → R is defined by Clearly, ρ 0 (p) = 1 for all p ∈ C. Also, the lapse function satisfies the Jacobi equation is minimal for all |t| < δ.
The last assertion follows from Lemma 2.1 and Σ t be stable.

Proof of Theorem 1.1
From Definition 1.1 and Lemma 2.2, there exists δ > 0 so that Σ t is a complete minimal stable surface, which is flat, totally geodesic and S = 0 along Σ t , for each |t| < δ. Now, we follows ideas of [1]. Since Ric(N t )+|A t | 2 ≡ 0 and H(t) = 0 for each |t| < δ, from (2.1) and Σ t being parabolic, we obtain that ρ t is constant. Thus, since Σ t is totally geodesic, is parallel. Also, the flow of N t is a unit speed geodesic flow (see [7]). Moreover, the map is a local isometry onto U = |t|<δ Σ t . Therefore, Φ is a diffeomorphism onto U, which implies that Y : C × (−δ, δ) → U is a globally defined unit Killing vector field. This implies that U is locally isometric to C × (−δ, δ). Now, assume that any stable minimal complete cylinder bifurcates for an uniform δ > 0. Then, we can start with a complete stable minimal cylinder Σ 0 that bifurcates, and then by the above considerations, Σ t , for each |t| < δ, is complete, flat, totally geodesic and S vanishes along Σ t . Moreover, Σ t is strongly stable for each |t| < δ. Note that Σ δ is a strongly stable minimal surfaces conformally equivalent to a cylinder, since it is limit of strongly stable minimal surfaces Σ t which are flat and totally geodesic, then Σ δ is totally geodesic, flat and S = 0 along Σ δ . Therefore, by Definition 1.1 and Lemma 2.2, there exists δ > 0 so that Σ t , −δ < t < 2δ, is flat, totally geodesic and S vanishes along Σ t . Continuing this argument, Σ t is flat, totally geodesic and S vanishes along Σ t for each t ∈ I, where I = R or I = S 1 .
As we did above, since Ric(N t ) + |A t | 2 ≡ 0 and H(t) = 0 for each t ∈ I, from (2.1) and Σ t being parabolic, we obtain that ρ t is constant. Thus, since Σ t is totally geodesic, is parallel, where I = R or I = S 1 . Also, the flow of N t is a unit speed geodesic flow (see [7]). Moreover, the map is a local isometry, which implies that it is a covering map. Therefore, Φ is a diffeomorphism, which implies that Y : C × I → M is a globally defined unit Killing vector field. This implies that M is locally isometric either to S 1 × R 2 or T 2 × R (here T 2 denotes the flat tori).