A splitting theorem for scalar curvature

We show that a Riemannian $3$-manifold with non-negative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.


Introduction
Let (M, g) be a connected, orientable, complete Riemannian 3-manifold with non-negative scalar curvature. D. Fischer-Colbrie and R. Schoen show in [10] that a connected, orientable, complete stable minimal immersion into (M, g) is conformal to a plane, a sphere, a torus, or a cylinder. They conjecture that (M, g) is flat if the immersion is conformal to the cylinder; cf. Remark 4 in [10]. M. Cai and G. Galloway point out a counterexample obtained from flattening standard R × S 2 near R × {great circle} in their concluding remark in [7]. They ask if the conjecture holds under the additional assumption that the immersion be "suitably" area-minimizing. In this paper, we prove the following result: Theorem 1.1. Let (M, g) be a connected, orientable, complete Riemannian 3-manifold with nonnegative scalar curvature. Assume that (M, g) contains a properly embedded surface S ⊂ M that is both homeomorphic to the cylinder and absolutely area-minimizing. Then (M, g) is flat. In fact, a cover of (M, g) is isometric to standard S 1 × R 2 upon scaling.
Note that this result is in satisfying analogy with the classical splitting theorem of J. Cheeger and D. Gromoll [9] in dimension 3, where scalar curvature takes the place of Ricci curvature and where area-minimizing cylinders stand in for length-minimizing geodesic lines. 1 Theorem 1.1 follows from the work of M. Anderson and L. Rodríguez [2] when we impose the much stronger assumption of bounded, non-negative Ricci curvature. The strategy of M. Anderson and L. Rodríguez [2] has been refined by G. Liu [13] to classify complete, non-compact Riemannian 3-manifolds with non-negative Ricci curvature. These ideas have been developed by the first-and second-named authors to establish the following scalar-curvature rigidity result for asymptotically flat 3-manifolds which had been conjectured by R. Schoen. 8]). The only asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that admits a non-compact, area-minimizing boundary is flat R 3 .
Our proof of Theorem 1.1 in this paper is a further development of these ideas. We now discuss challenges in the proof that are not present in the previously discussed works.
The goal is to construct a foliation of (M, g) by area-minimizing cylinders. The leaves of this foliation arise as limits of solutions of certain Plateau problems for least area. A major challenge we face here that has no substantial analogue in the proof of Theorem 1.2 is the a priori possible appearance of stable minimal planes or spheres (rather than cylinders or tori) in these limits. This scenario is addressed in Figure 2 below. The ambient scalar curvature may well be positive along such surfaces, as the examples in Remark 3 of [10] show.
We need an approach that is sensitive to the topology of solutions of the Plateau problems in the construction. At the same time, we have to make sure that these solutions pass to limits in a reasonable way. Recall from e.g. [21] that stable minimal surfaces in Riemannian 3-manifolds satisfy local curvature estimates that are independent of area bounds. In particular, sequences of such surfaces admit subsequential limits as pointed immersions. If each surface in the sequence is an area-minimizing boundary, then so is the limit. (If a small ambient ball intersects such a surface in two components, then these sheets have opposite orientation and are almost parallel. This scenario can be ruled out by a cut-and-paste argument.) However, limits of general areaminimizing surfaces can exhibit much greater complexity -think of condensing closed geodesics on the torus and compare with Remark A.1 below.
As such, the use of solutions of Plateau problems in the class of all (oriented) competitors risks the loss of local area bounds in the limit. On the other hand, the use of solutions in the class of boundaries risks the appearance of planes or spheres in the limit. These threats taken together force us to select the various classes of surfaces considered in the proof of Theorem 1.1 with great care.
We review the notions of area-minimizing surfaces that are used in this paper in Appendix A.
Subsequent to the paper of M. Cai and G. Galloway [7], there have been many further works establishing scalar curvature rigidity results in the presence of compact area-minimizing surfaces, including [6,4,5,11,14,15,19,1,16,18]. We anticipate that the techniques developed here lead to alternative proofs of these results. We plan to explore this possibility in forthcoming work.
Finally, we mention that parts of the strategy of M. Anderson and L. Rodríguez [2] depend on the assumption of non-negative Ricci curvature in a subtle but essential way. In particular, we do not see how to carry over the crucial area estimate [2, (1.5)] to the non-negative scalar curvature setting.

Tools
The following result is due to R. Schoen and S.-T. Yau [22,Section 5], except for the assertion that ϕ * g is flat in the latter alternative, which is due to D. Fischer-Colbrie and R. Schoen [10,Theorem 3].
Lemma 2.1. Let (M, g) be a Riemannian 3-manifold. Let ϕ : S → M be an orientable, complete, two-sided stable minimal immersion of a closed surface S such that R • ϕ ≥ 0, where R is the scalar curvature of (M, g). Then S is topologically either a sphere or a torus. In the latter alternative, the immersion is totally geodesic, the induced metric ϕ * g on S is flat, and R • ϕ = 0.
The following result due to D. Fischer-Colbrie and R. Schoen is part of Theorem 3 in [10].
Lemma 2.2. Let (M, g) be a Riemannian 3-manifold. Let ϕ : S → M be an orientable, complete, non-compact, two-sided stable minimal immersion such that R • ϕ ≥ 0 where R is the scalar curvature of (M, g). Then S with the induced metric ϕ * g is conformal to either the plane or the cylinder.
We refer to [23,17,3,20] as well as Appendix C of [8] for discussions and proofs of the following refinement of the latter alternative in Lemma 2.2.
where R is the scalar curvature of (M, g). If S is a cylinder, then the immersion is totally geodesic, the induced metric ϕ * g is flat, and R • ϕ = 0.
The following rigidity result is due to M. Cai and G. Galloway [7, Theorem 2]. Lemma 2.4. Let (M, g) be a connected, complete Riemannian 3-manifold with possibly empty weakly mean-convex boundary. We also assume that (M, g) has non-negative scalar curvature. If (M, g) contains a two-sided torus that has least area in its isotopy class, then (M, g) is flat.
Finally, we will need the following lifting property for absolutely area-minimizing surfaces.
Lemma 2.5. Let (M, g) be an orientable, complete Riemannian 3-manifold and let S ⊂ M be a properly embedded, orientable, absolutely area-minimizing surface without boundary. There is a covering p : The inclusion map lifts to a proper embeddingĩ : S →M andĩ * : π 1 (S) → π 1 (M ) is surjective. Moreover,S =ĩ(S) ⊂M is absolutely area-minimizing in (M ,g) whereg = p * g.
Proof. The asserted existence of covering and lift are standard, see Propositions 1.36 and 1.33 in [12]. To see thatS is area-minimizing, note that p is injective alongS and that every (properly embedded) competing surface forS projects to a properly immersed, competing surface for S. Using cut-and-paste arguments with small changes in area, we obtain a properly embedded competitor downstairs.

Proof of Theorem 1.1
Let S ⊂ M be a properly embedded cylinder that is absolutely area-minimizing in (M, g). In view of Lemma 2.5, we may assume that the inclusion map i : S → M induces a surjection i * : π 1 (S) → π 1 (M ). It follows from Lemma 2.3 that S ⊂ M is intrinsically flat. By scaling (M, g) if necessary, we may assume that S ⊂ M is isometric to standard S 1 × R.
If S ⊂ M is separating, then M \S has two components. We cut M along S and make a choice of component to obtain a connected, complete Riemannian 3-manifold whose boundary is connected. If S ⊂ M doesn't separate, then M \ S is connected. We cut M along S to obtain a connected, complete Riemannian 3-manifold whose boundary has exactly two components, of which we choose one. Either way, we denote the new Riemannian 3-manifold by (M ,ĝ) and the chosen component of its boundary by Σ. Note that Σ ⊂M is isometric to S and absolutely area-minimizing in (M ,ĝ). We denote the closed curve on Σ that corresponds to S 1 × {0} by γ. We write Σ h for the portion of Σ corresponding to Fix a unit speed geodesic c : [0, ε) →M with c(0) ∈ γ andċ(0) ⊥ T c(0) Σ. As in the proof of Theorem 1.2 in Appendix J of [8], we find a family of smooth Riemannian metrics {ĝ(r, t)} r,t∈(0,ε) onM with the following properties (illustrated in Figure 1): (i)ĝ(r, t) →ĝ in C 3 as t, r ց 0; (ii)ĝ(r, t) →ĝ smoothly as t ց 0 for r ∈ (0, ε) fixed; (iii)ĝ(r, t) =ĝ on {x ∈M : distĝ(x, c(2r)) ≥ 3r}; (iv)ĝ(r, t) <ĝ as quadratic forms in {x ∈M : distĝ(x, c(2r)) < 3r}; (v)ĝ(r, t) has positive scalar curvature in {x ∈M : r < distĝ(x, c(2r)) < 3r}; (vi)M is weakly mean-convex with respect toĝ(r, t).  where δ ∈ (0, 1) is chosen to satisfy (1) below. Among all compact, oriented surfaces in B h with boundary ∂Σ h that bound an open subset ofM , there is one whose area with respect toĝ(r, t, h) is least. Choose one such area-minimizing surface and denote it by Σ h (r, t). We claim that Σ h (r, t) intersects {x ∈M : distĝ(x, c(2r)) < 3r}. For if not, we have thatĝ =ĝ(r, t) along Σ h (r, t) and we may compute, as in [8, (12)], that This is a contradiction if we choose δ = δ(r, t, h) > 0 with A comparison with small geodesic spheres gives local area bounds for the surfaces Σ h (r, t) that are independent of all parameters; cf. Remark A.1. Using standard results in geometric measure theory, we may pass these surfaces to a subsequential limit as h → ∞ to obtain a properly embedded surface Σ(r, t). Note that Σ(r, t) is a boundary inM . It follows from the construction that this boundary is homologically* area-minimizing 2 with respect to g(r, t).
When r, t > 0 are sufficiently small, the surface Σ(r, t) contains a closed curve γ(r, t) that intersects {x ∈M : distĝ(x, c(2r)) ≤ 3r} and which is close to γ. Indeed, there is a small ball with center at c(0) ∈ γ where a piece of Σ(r, t) appears as a graph above T c(0) Σ. The geometric Harnack principle allows us to continue this piece of Σ(r, t) into a ribbon as we traverse γ. The ribbon must close up as we travel around γ, as otherwise there would be two nearby sheets -a scenario that contradicts the area-minimizing property of Σ(r, t). The componentΣ(r, t) of Σ(r, t) that contains this curve γ(r, t) converges to Σ as r, t ց 0. Observe thatΣ(r, t) is a boundary inM that is homologically* area-minimizing with respect toĝ(r, t); cf. Lemma 33.4 in [24].
Σ Figure 2. The possibility thatΣ(r, t) is a plane for r, t > 0 small enough can be ruled out by comparing the areas of the disks ∆(r, t), or by incompressibility of Σ.
We see from (v), Lemma 2.1, Lemma 2.2, and Lemma 2.3 that, for r, t > 0 small enough,Σ(r, t) either intersects {x ∈M : distĝ(x, c(2r)) ≤ r}, orΣ(r, t) intersects {x ∈M : distĝ(x, c(2r)) = 3r} but not {x ∈M : distĝ(x, c(2r)) < 3r}. Fix r > 0 small and pass to a geometric subsequential limit as t ց 0. We obtain a properly embedded boundary Σ(r) ⊂M which has a connected component Σ(r) that intersects {x ∈M : distĝ(x, c(2r)) ≤ 3r} and which is disjoint from Σ. As before, we see thatΣ(r) is a properly embedded boundary inM that is homologically* area-minimizing with respect toĝ. Clearly,Σ(r) is disjoint from Σ and contains a closed embedded curve γ(r) close to γ. The argument of the preceding paragraph shows that when r > 0 is small enough, the surfacê Σ(r) is diffeomorphic to either a torus or a cylinder. IfΣ(r) is a torus, then (M ,ĝ) is isometric to either standard S 1 × R × [0, ∞) or standard S 1 × R × [0, a] for some a > 0 by Lemma 2.4. We may thus assume thatΣ(r) is cylindrical for all r > 0 small. By Lemma 2.3,Σ(r) ⊂M is intrinsically flat, totally geodesic, and the ambient Ricci tensor evaluated in the normal direction vanishes alongΣ(r). Note thatΣ(r) converges to Σ as r ց 0. We now argue exactly as in the proof of [8, Theorem 1.6] to show that the ambient Riemann tensor vanishes along Σ.
We may repeat the above argument starting with any of the surfacesΣ(r) for r > 0 sufficiently small, using that they are homologically* area-minimizing in (M ,ĝ). 3 A continuity argument then gives that (M ,ĝ) is either standard S 1 × R × [0, ∞) or standard S 1 × R × [0, a] for some a > 0.
Lemma 3.1. Assumptions and notation as in the proof of Theorem 1.1 above. If the inclusion Σ ⊂M induces a trivial map π 1 (Σ) → π 1 (M ), then every connected, closed surface inM is separating.
Proof. It follows from the hypothesis and the construction ofM from M \ S that M is simply connected. (Here we are use that the map π 1 (S) → π 1 (M ) induced by the inclusion S ⊂ M is surjective.) Standard intersection theory gives that every connected, closed embedded surface N ⊂ M separates M . Indeed, if we assume that N ⊂ M is not separating, there is a closed embedded curve in M that intersects N transversely and exactly once. Such a curve cannot be homotopically trivial. Assume now thatN ⊂M is a connected and closed surface disjoint from the boundary ofM . It corresponds to a connected and closed surface N ⊂ M that is disjoint from S. Using that N separates M and the construction ofM from M , we conclude thatN separates.
A variation of this proof of Theorem 1.1 gives the following result.
Theorem 3.2. Let (M, g) be a connected, orientable, complete Riemannian 3-manifold with nonnegative scalar curvature. Assume that (M, g) contains a properly embedded, incompressible 4 surface S ⊂ M that is homeomorphic to the cylinder and homologically area-minimizing. Then (M, g) is flat. In fact, a cover of (M, g) is isometric to standard S 1 × R 2 upon scaling.
Proof. We follow the proof of Theorem 1.1 above, except for the following changes: in the standard cylinder S 1 × R. γ 1 is absolutely length-minimizing, γ 2 is not. Both γ 1 and γ 2 are homologically length-minimizing. γ 1 is homologically* length-minimizing, γ 2 is not.
Σ(r 1 , t 1 ) Σ(r 2 , t 2 ) Σ Figure 3. This figure shows a hypothetical scenario if we had not passed to the cover (using Lemma 2.5). We want to compare the areas ofΣ(r 1 , t 1 ) andΣ(r 2 , t 2 ) in a bounded set, using their respective minimizing properties. However, due to the presence of a neck, the surfaces are neither homologically nor homologically* related.
We conclude by describing how this non-standard area-minimizing property is tailored to a delicate aspect in the proof of Theorem 1.1. As we have just discussed, we cannot expect local area bounds for sequences of absolutely area-minimizing surfaces. For this reason, we want to fix the homology class of the surfaces we work with, or at least keep it in check. If we work with homological area-minimizers, it is not clear how to rule out minimizing planes or spheres in the proof of Theorem 1.1 by cut-and-paste arguments. Figure 3 shows a scenario where the planes we worry about are not homologically related. To deal with this scenario, we pass to the cover using Lemma 2.5 at the beginning of the proof. The situation after passing to the cover is shown in Σ(r 1 , t 1 ) Σ(r 2 , t 2 ) Σ . . . . . . Figure 4. A depiction of the situation in Figure 3 after passing to the appropriate cover (using Lemma 2.5). Now, compact pieces ofΣ(r 1 , t 1 ) andΣ(r 2 , t 2 ) are homologically* but not homologically comparable. This is because they differ by an open set (shaded grey) which is necessarily unbounded. Figure 4. Now, the surfaces whose areas we want to compare bound a non-compact region. They are homologically* related, but not homologically related.