title: Generalized soap bubbles and the topology of manifolds with positive scalar curvature
authors: Otis Chodosh Chao Li
text: 
  
  Soap bubbles and positive scalar curvature]Generalized soap bubbles and the topology of manifolds with positive scalar curvature
  
  OC: Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA
  ochodosh@stanford.edu
  
  CL: Department of Mathematics, Princeton University, Fine Hall, 304 Washington Road, Princeton, NJ 08540, USA
  Current Address: Courant Institute, New York University, 251 Mercer St, New York, NY 10012, USA
  chaoli@nyu.edu
  
  
  
  
  
  We prove that for n∈{4,5} a closed aspherical n-manifold does not admit a Riemannian metric with positive scalar curvature. 
  
  
  Additionally, we show that for n≤ 7, the connected sum of a n-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with forthcoming contributions by Lesourd–Unger–Yau, this proves that the Schoen–Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature.
  
  A key geometric tool in these results are generalized soap bubbles—surfaces that are stationary for prescribed-mean-curvature functionals (also called μ-bubbles). 
  
  
  [
      Chao Li
      Aug. 2020
  =============
  
  
  
  
  
  § INTRODUCTION
  
  
  We begin by recalling the following well-known result proven by R. Schoen and S.-T. Yau and by M. Gromov and B. Lawson.[D. Stern has recently discovered an interesting new proof of this when n=3 (see <cit.>).] 
  [Geroch Conjecture <cit.>]
  The n-torus does not admit a Riemannian metric of positive scalar curvature. 
  
   This result (and generalizations thereof) has had several important consequences, including Schoen–Yau's proof of the positive mass theorem in general relativity <cit.> and Schoen's resolution of the Yamabe problem concerning conformal deformations to constant scalar curvature <cit.>. 
  
  In this paper, we provide two extensions of Theorem <ref>, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5, as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. 
  
  We prove these results by analyzing stable solutions to the prescribed mean curvature problem (called here μ-bubbles). This seems to be the first use of these surfaces to yield global topological restrictions on positive scalar curvature; in Section <ref> below, we discuss previous applications of μ-bubbles for local geometry of scalar curvature (e.g. comparison theorems). We expect that μ-bubbles will find other topological applications. 
  
  
  
  
   §.§ Aspherical manifolds
   Recall that a smooth manifold is aspherical if it has contractible universal cover.  For example, any closed manifold covered by a Cartan–Hadamard manifold (such as the torus) is aspherical. Our first main result is as follows.
  
  For n ∈{4,5}, a smooth closed aspherical n-manifold N does not admit a smooth Riemannian metric with positive scalar curvature. Any metric of non-negative scalar curvature on N is flat. 
  
  
  
  In a 1987 survey article <cit.>, Schoen–Yau first proposed that all closed aspherical manifolds do not admit metrics of positive scalar curvature.[S.-T. Yau has informed us that they in fact conjectured this in the early 1980s.] They outlined a proof of this in 4-dimensions <cit.>, but many parts of the proof have not been given. Resolving these missing parts is essential to carry out their outline, as we do in this paper. Furthermore, the case of 5-dimensional manifolds is considerably more involved. See Section 1.3 for a more detailed discussion.
  
  Theorem <ref> (in all dimensions n≥ 4) was also conjectured in a slightly different form by M. Gromov (see <cit.>). It has some link with the Novikov conjecture on topological invariance of certain polynomials of Pontryagin classes, as explained in <cit.>. Furthermore, as discussed in <cit.>, Theorem <ref> (and its conjectural higher dimensional analogue) is one of the central questions in the study of geometric and topological properties of manifolds with positive scalar curvature. 
  
  
  Recently, J. Wang proved Theorem <ref> in the special case that N is of dimension 4 with nonzero first Betti number <cit.>.[As pointed out in <cit.>, there exist infinitely many aspherical 4-manifolds with zero first Betti number <cit.>. ] 
  
  We note that in the first version of this article, we only considered Theorem <ref> in the case of n=4. As we were finishing writing down the generalization to n=5, we received a preprint from M. Gromov <cit.> containing a proof of the same generalization, as well as further extensions to certain non-compact 5-dimensional manifolds. Both approaches seem relatively similar, but certain central technical steps (compare Section 6.3 and 6.4 in this paper to <cit.>) are obtained via different methods (and were obtained independently). 
  
  
  
  
  
  
  
  
  
  
   §.§ The Geroch conjecture with arbitrary ends
   Our second main result resolves a question arising in the work of Schoen–Yau on locally conformally flat manifolds <cit.> (cf. <cit.>).
  
  
  Let n≤ 7. For any n-manifold X, the connect sum T^n# X does not admit a complete metric of positive scalar curvature. The only complete metric of non-negative scalar curvature on T^n#X is flat. 
  
  
  For example, this implies that a punctured torus does not admit a complete metric of positive scalar curvature (note however, that this particular statement follows from work of Gromov-Lawson in <cit.> based on relative index theorems). When X is compact, Theorem <ref> is well-known: it has recently been proven in all dimensions by Schoen–Yau <cit.> via an analysis of singular minimal surfaces (cf. <cit.>). 
  
  We emphasize that M. Lesourd, R. Unger, and S.-T. Yau have recently proven <cit.> Theorem <ref> for n=3. Their proof extends to 3<n≤ 7 under certain technical assumptions on the geometry and topology of the manifold at infinity. Our proof is different at a technical level (even when n=3) as compared to theirs (which involves analyzing non-compact stable minimal hypersurfaces).
  
  
  The main reason to consider Theorem <ref> comes from the study of the topology of locally conformally flat manifolds with nonnegative scalar curvature. In their foundational work on these manifolds <cit.> (cf. <cit.>), Schoen–Yau have proven that the geometry of the developing map of a locally conformally flat manifold with non-negative scalar curvature is strongly influenced by the mass of the manifold obtained via a Green's function conformal blown-up (motivated by Schoen's solution to the Yamabe problem <cit.>). Such a blown-up manifold will have a distinguished asymptotically flat end, but will also have other ends with metrics that are complete but do not have any other constraints on their geometry or topology. 
  
  
  If there were no such uncontrolled ends, a well-known argument due to Lohkamp allows one to reduce the positive mass theorem to the Geroch conjecture for T^n # M (where M is compact). Such a reduction is delicate for asymptotically flat manifolds with other complete ends, since the geometry along the other ends could affect certain global arguments. However, Lesourd–Unger–Yau have recently proven <cit.> that, through a careful analysis of the Green's function and modification of Lohkamp's argument, one can reduce[For completeness, we outline this reduction in the appendix of this paper.] the study of the manifolds arising in the work of Schoen–Yau <cit.> to the situation in Theorem <ref>. As such, by combining Theorem <ref> with results in <cit.> and <cit.>, we have the following definitive result. 
  
  Suppose that (M^n,g) is a complete Riemannian manifold with R_g≥ 0. If Φ : M → S^n is a conformal map, then Φ is injective and ∂Φ(M) has zero Newtonian capacity. 
  
  As shown in <cit.>, the Liouville result in Corollary <ref> has strong consequences for the higher homotopy groups of locally conformally flat manifolds with nonnegative scalar curvature. 
  
  
  Under additional assumptions on the geometry or dimension of M (including, e.g., that n≥ 7), this was proven by Schoen–Yau <cit.>. In fact, Corollary <ref> was announced by Schoen–Yau in full generality <cit.> (cf. <cit.>), but the essential ingredient along the lines of Theorem <ref> has never appeared. Finally, we reiterate that Lesourd–Unger–Yau <cit.> have proven Corollary <ref> when n=3 (as well as for 4≤ n≤ 7 with certain additional assumptions on the geometry and topology of (M,g) at infinity). 
  
  
  
  
  
  We remark that our proof of Theorem <ref> can be generalized to allow certain other manifolds in place of the torus, see Section <ref> and <cit.>. We expect that one may remove the dimensional restriction (following <cit.>) but we do not pursue this here.
  
  
  
  
   §.§ Idea of the proofs of Theorem <ref> and Theorem <ref>
  
  
  We explain the idea of the proofs of the results described above. As will be seen later, both Theorem <ref> and Theorem <ref> have the same central difficulty: it is hard to find a (compact) stable minimal surface due to non-compactness of the ambient manifold. Thus, we will mostly focus on Theorem <ref> here, and briefly indicate the strategy of Theorem <ref> at the end.
  
  Our strategy of proof of Theorem <ref> extends the outline of Schoen–Yau in <cit.> when n=4. We now describe their outline (of Theorem <ref> when n=4) and subsequently detail our contributions as well as how to generalize the strategy to n=5. 
  
  
  
  	Suppose to the contrary, that (N^4,g) is a closed aspherical manifold with R(g)≥ 1. Pass to a non-compact cover (N,g) which is homotopy equivalent to S^1. Fix [σ] a generator of π_1(N). Minimize area among hypersurfaces dual to [σ] to find a complete[In our paper, we find it convenient to instead work with a stable minimal hypersurface with fixed boundary that is very far from σ, but this is not an important point.] stable minimal hypersurface M.  By Schoen–Yau's inductive descent technique, the stable minimal hypersurface (M,g|_M) is Yamabe positive. 
  	
  	Schoen–Yau suggest that one can derive a contradiction from this as follows.  Suppose that one could find a large region Ω⊂ M so that each component of ∂Ω (there might be many) has controlled area. One can hope that each component of ∂Ω can be filled in by some 3-manifold in N of bounded diameter. The existence of such a fill-in is expected here, as N is a covering space of a closed manifold (cf. the Lemma below Theorem 6 in <cit.>). Given this, it would be possible to fill in each component of ∂Ω without affecting the intersection with σ. This would yield a 3-cycle in N with non-trivial algebraic intersection with σ. This would contradict H_3(N) =0, finishing the proof. 
  
  
  
  
  
  In this paper, we obtain a complete proof of Theorem <ref> (when n=4) along the lines of the outline given above. Our main contributions are the resolutions of the following fundamental difficulty arising in the outline above: to actually find Ω as stated. One intuitive way to approach this would be to exploit the fact that stable minimal surfaces in a three manifold with scalar curvature ≥ 2 are spherical and have area ≤ 4π (see <cit.> for a related rigidity result). However, there is no reason that M will contain any minimal surfaces, much less admit an exhaustion by regions bounded by stable minimal surfaces. 
  
  Instead, we rely on generalized soap bubbles—surfaces that are stationary for the prescribed-mean-curvature functional (called here the μ-bubble after <cit.>). The use of this functional in scalar curvature problems was first considered by Gromov in <cit.>. Recently, this approach was used by Gromov to obtain a list of comparison theorems in scalar curvature <cit.>, and later expanded in work of J. Zhu <cit.>.[The prescribed-mean-curvature functional has recently been considered in other geometric problems as well, cf. <cit.>.] By applying this technique, we can localize the minimizer while still retaining the geometric properties used in the inductive descent argument. 
  
  
  
  
  We also note that an additional issue with the outline as explained above is that the geometry of (M,g  |_M) after conformally deforming to positive scalar curvature might be dramatically different near infinity from the geometry of (N,g). To handle this, we combine the μ-bubble method with the warped product descent technique from <cit.>. 
  
  
  Next, we briefly discuss the generalization of this argument to the case of n=5. We argue similarly to the above sketch and produce a curve σ and 4-dimensional minimal surface M_4  (we find it convenient to work in the universal cover and with a minimizing line σ, but this is not an important issue) with boundary far from σ, so that M_4 has non-trivial algebraic intersection with σ. We can then use the μ-bubble argument as in n=3 to find a region in M_4 whose boundary is Yamabe positive and lies far away from σ. At this point, we encounter a serious difficulty not present in the previous situation: 3-manifolds with positive scalar curvature can have arbitrarily large diameter. 
  
  As such, we develop a slice-and-dice method that we use to decompose the 3-dimensional μ-bubble M_3 into pieces of controlled diameter and overlap. To do so, we first slice the μ-bubble by 2-dimensional spheres chosen to simplify H_2(M_3) appropriately. Then, we dice the resulting manifold with boundary, using free boundary μ-bubbles (if we did not slice first, the μ-bubble argument would only produce a decomposition into regions with bounded distance to the boundary, but if there are many boundary components, such a region could have arbitrarily large diameter). Because we arrange that the free boundary μ-bubbles are of disk type, this dices M_3 into connected components of controlled diameter, and so that the overlap of the sets (along their boundaries) has controlled diameter. This is sufficient to complete the fill-in argument explained for n=4 above.
  
  We note that the slice-and-dice method can be used to show that a compact 3-manifold with positive scalar curvature can be mapped to a (1-dimensional) graph by a map with pre-images of uniformly bounded diameter, or equivalently that such a manifold has bounded first Uryson width. See also <cit.> for related results. Y. Liokumovich and D. Maximo have recently obtained a related (but strictly more general) result along these lines; it would be interesting to compare the two approaches.  
  
  
  
  Finally we briefly motivate the proof of Theorem <ref>. We again seek a μ-bubble representing a suitable homology class and apply the Schoen-Yau descent argument (in fact, we argue slightly differently to this, by passing to a certain covering space so that we can consider boundaries rather than arbitrary homological relations). The key point in this argument is then to find the correct prescribing mean curvature function. 
  
  A basic difficulty present here (as compared to the previous discussion) is that we are only assuming positivity of the scalar curvature rather than uniformly positive scalar curvature. The standard μ-bubble technique (like the one we use in Theorem <ref>) could potentially fail if—for example—the scalar curvature decayed faster than quadratically. However, because there is a sufficiently nice covering space, we can make the argument work (morally, this has to do with the fact that the torus has sufficiently complicated topology). We note that a similar argument (cf. Theorem <ref> and <cit.>) extends Theorem <ref> to manifolds in the form M # X, where M is a Schoen–Yau–Schick manifold (see <cit.>) and X is arbitrary.  
  
  
  The paper is organized as follows. We prove Theorem <ref> in Section 2-6. In Section <ref>, we provide some topological preliminaries concerning aspherical n-manifolds. In Section <ref> we discuss μ-bubbles. and in Section <ref> we discuss the free boundary version of μ-bubbles. In Section <ref> we prove diameter bounds that will apply to stable 2-dimensional μ-bubbles (with  free boundary or not). Then in Section <ref> we prove Theorem <ref> by the slice-and-dice method mentioned above. We prove Theorem <ref> in Section <ref>. In Section <ref>, we discuss an extension of Theorem <ref> to Schoen–Yau–Schick manifolds. 
  
  
  The authors were informed by X. Zhou that recently, S.-T. Yau and X. Zhou made some progress (independent of this paper) towards completing the Schoen–Yau outline of Theorem <ref> in dimension n=4.
  
  
  
   §.§ Acknowledgements
   We thank the referees for their helpful suggestions. We are grateful to Brian White useful discussions about isoperimetric inequalities, and to Boyu Zhang for several helpful conversations about 4-manifold topology. We would like to thank Christos Mantoulidis, Davi Maximo, Misha Gromov, and Fernando Codá Marques for their interest as well as Richard Schoen for bringing the K(π,1) question to our attention and for his constant encouragement. 
  
  We are grateful to Martin Lesourd for answering several questions concerning their work <cit.>. Finally, the second named author wants to thank Xin Zhou for sharing his insights on the Schoen-Yau survey paper <cit.> in the early stages of this work.
  
  
  O.C  was supported in part by a Terman Fellowship, a Sloan Fellowship, NSF grants DMS-1811059/2016403 and 2304432. C.L. was supported by NSF grant DMS-2005287. 
  
  
  
  
  
  
  § TOPOLOGICAL PRELIMINARIES
  
  
  In this section we collect some basic topological facts on closed aspherical manifolds. Let N be a smooth connected n-manifold. N is called aspherical, if its homotopy groups π_j(N) is trivial for all integers j > 1. Equivalently, N is a Eilenberg–MacLane space K(π,1), where π is the fundamental group of N. It is a standard result that N is aspherical if and only if its universal cover, Ñ, is contractible <cit.>. 
  
  We fix (N,g) a compact aspherical Riemannian n-manifold and denote by Ñ its universal cover. The following is standard. 
  
  Ñ is non-compact.
  
  
  Because Ñ is contractible H_n(Ñ,_2) = 0. However, any compact n-manifold has H_n(Ñ,_2) = _2.
  
  
  The next lemma is well known (and holds for the universal cover of any compact manifold). We recall the proof here for completeness. 
  
  
  There exists a geodesic line σ : → (Ñ,g) , i.e., d(σ(t_1),σ(t_2)) = |t_2-t_1| for all t_1,t_2∈. 
  
  
  Fix p ∈Ñ, and choose p_i ∈Ñ diverging. Let σ_i denote minimizing geodesics between p and p_i. Passing to a subsequence, σ_i converge to σ' : [0,∞) →Ñ a minimizing ray. For t_i→∞ choose deck transformations Φ_i so that d(p,Φ_i(σ'(t_i))) is uniformly bounded. Then, σ_i'(t) = Φ_i(σ'(t+t_i)) subsequentially converges to a geodesic line σ.
  
  
  Choose two smooth functions ρ_1,ρ_2 : N→ so that 
  
      |ρ_1(p) - d(p,σ([0,∞))| ≤ 1,      |ρ_2(p) - d(p,σ(0))| ≤ 1.
  
  For a large regular value of ρ_1, L_1≫1, set U : = ρ_1^-1((-∞,L_1)). By construction, ∂ U is a smooth properly embedded hypersurface. 
  
  
  For L_2 sufficiently large, σ∩∂ U ⊂ρ_2^-1((-∞,L_2]). 
  
  
  Observe that σ([0,∞]) ⊂ U.  Moreover, σ((-∞,-L_1-4]) ∩ U = ∅. Indeed, if t_1 < -L_1-4 has σ(t_1) ∈ U then 
  
      d(σ(t_1),σ([0,∞)) ≤ L_1 + 2
  
  As such, there is t_2 ≥ 0 with
  
      L_1+4 ≤ |t_1 - t_2| = d(σ(t_1),σ(t_2)) ≤ L_1 + 3.
  
  This is a contradiction. Thus σ∩∂ U is contained in a compact set. This completes the proof. 
  
  
  
  
  
  Fix a regular value L_2 ≫ L_1 of the function ρ_2|_∂ U and set 
  
      M = (∂ U) ∩ρ_2^-1((-∞,L_2)).
  
  Note that M is a smooth properly embedded compact oriented hypersurface with boundary. Perturb σ slightly so that it intersects M transversely (and σ∩∂ U = σ∩ M) and we still have
  
      |ρ_1(p) - d(p,σ([0,∞))| ≤ 2,      |ρ_2(p) - d(p,σ(0))| ≤ 2.
  
  We now verify several properties of M and σ, as illustrated by Figure <ref>.
  
  
  
  
  For L_2 sufficiently large, the curve σ has non-zero algebraic intersection with M.
  
  
  The curve σ(t) leaves and then re-enters U in (oppositely oriented) pairs until the smallest intersection time t, after which it never intersects ∂ U. Because σ∩∂ U = σ∩ M this proves the assertion. 
  
  
  
  
  For L_2 sufficiently large, ∂ M ≠∅ and d(∂ M,σ()) ≥ L_1 - 4. 
  
  
  Note that ∂ M ⊂ρ_2^-1(L_2). First, suppose that ∂ M = ∅. Then, by Lemma <ref>, we could conclude that [M] ≠ 0 ∈ H_n-1(Ñ) = 0. This is a contradiction. 
  
  Suppose there is t_2 ∈ and p∈∂ M with d(p,σ(t_2)) ≤ L_1 - 3. Suppose that t_2 ≥ 0. Then,
  
      ρ_1(p) ≤ d(p,σ([0,∞))) + 2 ≤ L_1 - 1.
  
  This cannot hold (since ∂ M ⊂ρ_1^-1(L_1)). As such, we see that t_2 < 0. Moreover, since d(∂ M,σ(0)) ≥ L_2 -2, we find
  
      |t_2| = d(σ(t_2),σ(0)) ≥ d(p,σ(0)) - d(p,σ(t_2)) ≥ L_2 - L_1 + 1.
  
  On the other hand, we know there must be t_1 ≥ 0 with d(p,σ(t_1)) ≤ L_1 + 2. We have
  
      L_2 - L_1 + 1 ≤ |t_1 - t_2| = d(σ(t_1),σ(t_2))  
      ≤ d(p,σ(t_1)) + d(p,σ(t_2)) ≤ 2L_1 - 1.
  
  This is a contradiction as long as L_2 > 3L_1 - 2.
  
  
  
  For r>0 there is R=R(r) with the following property. Suppose that α is a k-cycle in Ñ with α⊂ B_r(p) for some p∈Ñ. Then α = ∂β for β⊂ B_R(p). 
  
  
  Fix p_0∈Ñ. Since Ñ is contractible, for r>0, B_r(p_0) is contractible in B_R_0(r)(p_0) for some function R_0=R_0(r)<∞. In particular H_k(B_r(p_0)) → H_k(B_R_0(p_0)) is the zero map, for k>0. 
  
  For any p∈Ñ, there is a deck transformation Ψ so that d(p_0,Ψ(p)) ≤ N. As such, Ψ(B_r(p)) ⊂ B_r+ N(p_0). There is β∈ B_R_0(r+ N)(p_0) ⊂ B_R_0(r+ N)+ N(Ψ(p)) with ∂β = Ψ(α). As such 
  
      Ψ^-1(β) ⊂ B_R_0(r+ N)+ N(p)
  
  has ∂ (Ψ^-1(β)) = α. This completes the proof. 
  
  
  
  
  § WARPED Μ-BUBBLES
  
  In this section we recall general existence and stability results for warped μ-bubbles. For n≤ 7, consider (M,g) a Riemannian n-manifold with boundary and assume that ∂ M = ∂_- M ∪∂_+ M is a choice of labeling the components of ∂ M so that neither of the sets ∂_± M are empty. Fix a smooth function u > 0 on M and a smooth function h on M with h→±∞ on ∂_± M. 
  Choose a Caccioppoli set Ω_0 with smooth boundary ∂Ω_0 ⊂ M and ∂_+ M⊂Ω_0. 
  Consider the following functional
  
      (Ω)=∫_∂^* Ω u  d^n-1 - ∫_M (χ_Ω-χ_Ω_0)hu   d^n,
  
  for all Caccioppoli sets Ω in M with ΩΔΩ_0⋐ M. We will call Ω minimizing  in this class a μ-bubble. 
  
   Geometrically, this is equivalent to the functional
  
      (Ω× S^1)= ^n-1(∂^*(Ω× S^1))- ∫_Ω× S^1 (χ_Ω× S^1-χ_Ω_0× S^1)h̃  d^n,
  
  for S^1-invariant Caccioppoli sets Ω× S^1 inside (M_1× S^1,g̃), where g̃ is the warped product metric g̃=g+u^2 dt^2, and h̃ is defined on M× S^1 by h̃(x,t)=h(x). However, we find it simplest to work with the form (<ref>) instead of the warped product formulation. 
  
  
  
   §.§ Existence of μ-bubbles
    The existence and regularity of a minimizer of  among all Caccioppoli sets (without any equivarient assumptions) was claimed by Gromov in <cit.>, and was rigorously carried out by Zhu in <cit.>. For the sake of completeness, we include a proof here.
  
  There exists a smooth minimizer Ω for  such that ΩΔΩ_0 is compactly contained in the interior of M.
  
  
  	Let Ω be a Caccioppoli set in M such that ΩΔΩ_0 ⋐ M. By a standard approximation argument, we can assume that Ω has smooth boundary. We first show that, by adding to Ω a neighborhood of ∂_+ M, and subtracting from it a neighborhood of ∂_-M, one decreases _h. For τ>0, denote by Ω_±^τ the distance-τ neighborhood of ∂_± in M. Choosing τ sufficiently small, Ω_±^τ has a foliation {S_±^ρ}_ρ∈ [0,τ] by smooth equidistant hypersurfaces to ∂_± M. Denote by η the unit normal vector field of {S_±^ρ} defined in this Ω_±^τ, pointing into M along ∂_± M. Let τ>0 be sufficiently small so that 
  	
      h u >  (u η)  in Ω_+^τ,    h u < -  (uη)  in Ω_-^τ.
  
  	
  	We compute
  	
      (Ω∪Ω_+^τ)-(Ω)
         =∫_∂^*(Ω∪Ω_+^τ)u   d^n-1 - ∫_∂Ω u   d^n-1- ∫_Ω_+^τ∖Ω hu   d^n
         = ∫_∂Ω_+^τ∖Ω u   d^n-1- ∫_∂Ω∩Ω_+^τ u   d^n-1- ∫_Ω_+^τ∖Ω hu   d^n
          < ∫_∂Ω_+^τ∖Ω u   d^n-1- ∫_∂Ω∩Ω_+^τ u   d^n-1- ∫_Ω_+^τ∖Ω(u η)   d^n.
  
  Moreover,
  
      ∫_Ω_+^τ∖Ω(uη)   d^n   = ∫_∂Ω_+^τ∖Ω (η·ν) u   d^n-1 - ∫_∂Ω∩Ω_+^τ (η·ν) u   d^n-1
         ≥∫_∂Ω^τ_+∖Ω u   d^n-1 - ∫_∂Ω∖Ω^τ_+ u   d^n-1.
  
  	Thus (Ω∩Ω_+^τ)< (Ω). By an analogous calculation, (Ω∖Ω_-^τ)<(Ω). Hence it suffices to consider the infimum of  among , where  is the collection of Caccioppoli sets that contain Ω_+^τ and are disjoin from Ω_-^τ. Since |hu|<C_1 in M∖(Ω_+^τ∪Ω_-^τ), we conclude that (Ω)>-C_1 ^n(M) for all  Ω∈. Hence I=inf{(Ω): Ω∈} exists. Take a sequence Ω_k∈ with (Ω_k)→ I. Then ^n-1(∂Ω_k)<C(I+C_1^n(M)). By BV-compactness, taking a subsequence, the sets Ω_k converge to a Caccioppoli set Ω. It follows that Ω is a minimizer of , and thus has smooth boundary by standard regularity theory <cit.>.
  
  
  
  
   §.§ Stability
   We now discuss the first and second variation for a warped μ-bubble. 
  
  If Ω_t is a smooth 1-parameter family of regions with Ω_0 = Ω and normal speed ψ at t=0, then 
  
      t (Ω_t)=∫_Σ_t (Hu + ∇_M u, ν - hu)ψ  d^n-1
  
  where H is the scalar mean curvature of ∂Ω_t and ν is the outwards pointing unit normal. In particular, a μ-bubble Ω satisfies 
      H = -u^-1∇_M u,ν + h
  
  along ∂Ω. 
  
  
  
  Consider a μ-bubble Ω with ∂Ω = Σ. Assume that Ω_t is a smooth 1-parameter family of regions with Ω_0 = Ω and normal speed ψ at t=0, then (ψ):=d^2/dt^2|_t=0((Ω_t))≥ 0 where (ψ) satisfies 
  
      (ψ) 
         ≤∫_Σ(|∇_Σψ|^2 u -12 (R_M - 1 -R_Σ +|A_Σ|^2)ψ^2 u+ (Δ_M u  -Δ_Σ u)ψ^2     .
         .          - 12 u^-1∇_M u,ν^2ψ^2  -  12 (1 + h^2 + 2 ∇_M h,ν)ψ^2 u 
      )d^n-1.
  
  
  
  Differentiating the first variation, we find
  
      (ψ) 
         = ∫_Σ(-ψ u Δ_Σψ -  12 (R_M - R_Σ + |A_Σ|^2 + H_Σ^2)ψ^2 u  + H_Σ⟨∇_M u,ν⟩ψ^2  .
         .          +D^2u(ν,ν) ψ^2 - ⟨∇_Σ u,∇_Σψ⟩ψ- ⟨∇_M(hu), ν⟩ψ^2 ) d^n-1
         =∫_Σ(|∇_Σψ|^2 u -12 (R_M-R_Σ +|A_Σ|^2)ψ^2 u -34 H_Σ^2 ψ^2 u .
         .          + (Δ_M u  -Δ_Σ u)ψ^2 - ∇_M h,νuψ^2 - h∇_M u, νψ^2 )d^n-1.
  
  Now use 
  
      12 H_Σ^2 ψ^2 u =  12 u^-1∇_M u,ν^2 ψ^2  -h ∇_M u,νψ^2  +  12 h^2 ψ^2 u
  
  to write
  
      (ψ) 
         ≤∫_Σ(|∇_Σψ|^2 u -12 (R_M - 1 -R_Σ +|A_Σ|^2)ψ^2 u+ (Δ_M u  -Δ_Σ u)ψ^2     .
         .          - 12 u^-1∇_M u,ν^2ψ^2  -  12 (1 + h^2 + 2 ∇_M h,ν)ψ^2 u 
      )d^n-1.
  
  This completes the proof. 
  
  
  
  
  
  § FREE BOUNDARY WARPED Μ-BUBBLES
  
  
  We will need a generalization of the previous discussion to the free boundary case. For n≤ 7, suppose that (M^n,g) is a Riemannian n-manifold with co-dimension 2 corners in the sense that any point has a neighborhood diffeomorphic to one of the following: ^n, { x ∈^n : x^n≥ 0} or {x ∈^n : x^n-1,x^n ≥ 0}. 
  
  Assume that M∖ M = ∂_+M ∪∂_- M ∪∂_0 M where ∂_± M,∂_0 M are all smooth submanifolds of M (possibly with boundary). We assume that ∂_+ M ∩∂_- M = ∅ and ∂_± M ∩∂_0M consists of smooth co-dimension 2 closed submanifolds. Moreover, we assume that ∂_± M meets ∂_0M orthogonally. 
  
  For an open set Ω⊂ M we denote by ∂Ω the topological boundary in the sense that ∂Ω = Ω∩Ω^c. Observe that with this definition, if Ω =M, ∂Ω = ∅. 
  
  Fix a function u ∈ C^∞(M) with u>0 and h ∈ C^∞(M∖ (∂_+M∪∂_-M)). Assume that h→±∞ on ∂_± M. Consider a Caccioppoli set Ω_0 with smooth boundary satisfying ∂Ω_0 ⊂ M ∪∂_0 M. 
  
  Consider the functional (as before)
  
      (Ω) = ∫_∂^* Ω u   d^n-1 - ∫_M (χ_Ω - χ_Ω_0) hu   d^n
  
  for all Caccioppoli sets Ω in M with ΩΔΩ_0 ⋐ M∖ (∂_+M∪∂_-M). By similar arguments to the previous section, we can conclude the following result:
  
  There exists Ω with ∂Ω⊂ M ∪∂_0 M minimizing  among such regions. The boundary ∂Ω is smooth and meets ∂_0M orthogonally. We have
  
      H = - u^-1∇_M u,ν_∂Ω + h
  
  along ∂Ω. Finally, if Σ is a component of ∂Ω, then for any ψ∈ C^1(Σ), we have 
  
      0    ≤∫_Σ(|∇_Σψ|^2 u -12 (R_M - 1 -R_Σ +|A_Σ|^2)ψ^2 u+ (Δ_M u  -Δ_Σ u)ψ^2     .
         .          - 12 u^-1∇_M u,ν_Σ^2ψ^2  -  12 (1 + h^2 + 2 ∇_M h,ν_Σ)ψ^2 u 
      )d^n-1
              - ∫_∂Σ A_∂_0M (ν_Σ,ν_Σ) ψ^2 u   d^n-2 .
  
  
  
  Existence and interior regularity follow in a similar manner to Proposition <ref> with the following modification. To show that a minimizer Ω exists such that ΩΔΩ_0 is disjoint from ∂_± M, for τ>0 sufficiently small, we find a foliation {S^ρ_±}_ρ∈ [0,τ] of a small neighborhood Ω_±^τ of ∂_± M such that each S^ρ_± meets ∂_0 M orthogonally. Denote by η the unit normal vector field of {S_±^ρ} pointing into M along ∂_± M. Choose τ>0 sufficiently small so that
  
      hu> (uη)  in Ω_+^τ,    hu < - (uη)  in Ω_-^τ.
  
  Then a similar computation as in the proof of Proposition <ref> (the boundary term from the divergence theorem vanishes since each S_±^ρ meet ∂_0 M orthogonally) shows that we always have (Ω∩Ω_+^τ)<(Ω) and (Ω∖Ω_-^τ)<(Ω). In particular, this shows that the minimizer Ω is disjoint from the co-dimension 2 corners. Thus, regularity at the free boundary follows by modifying the regularity results for free boundary isoperimetric surfaces in <cit.> (see also <cit.>) so as to apply to the μ-bubble functional. The second variation follows as before, but with the additional boundary term arising exactly as in the calculation of the second variation of free boundary minimal surfaces. 
  
  
  
  
  
  
  § DIAMETER BOUNDS FOR CERTAIN SURFACES
   
  In this section we derive diameter bounds for certain surfaces (in practice these will be stable μ-bubbles in certain 3-manifolds). Schoen–Yau <cit.> have proven that stable minimal surfaces in 3-manifolds of positive scalar curvature satisfy such an inequality. Moreover, they have proven that the 1-dimensional components of their minimal k-slicings satisfy a length bound <cit.>. Since we will apply this slicing idea for μ-bubbles (possibly with boundary), we find it convenient to state and prove the following results, analogous to the Schoen-Yau estimates <cit.>. 
  
  We use μ-bubbles to give a slightly different proof of these diameter bounds as to compared to those in <cit.>. 
  
  
  For closed 2-dimensional Riemannian manifold (Σ^2,g), suppose there is a smooth function λ > 0 so that 
  
      Δ_Σλ≤ - (K_0-K_Σ) λ +  12 λ^-1 |∇_Σλ|^2
  
  for some K_0 ∈ (0,∞). Then _g Σ≤√(2K_0)π. 
  
  
  
  For contradiction, suppose there are p,q ∈Σ with 
  
      L : = d(p,q) >  L_0 : = √(2K_0)π.
  
  There is >0 and a smooth function 
  
      ρ : Σ∖ (B_(p)∪ B_(q)) → (-L_0/2,L_0/2)
  
  with |ρ| ≤ 1- and ρ→  -L_0/2 at ∂ B_(p) and ρ→ L_0/2 at ∂ B_(q). We then take 
  
      h(x) = - 2(1-) πL_0tan(πL_0ρ(x)).
  
  Observe that 
  
      K_0 +  12 h(x)^2 - |∇_Σ h|(x) 
         ≥ K_0 + 2(1-)^2 π^2L_0^2 ( tan^2(πL_0ρ(x)) - ^2(πL_0ρ(x)))
          = K_0 - 2(1-)^2 π^2L_0^2
          = K_0 - (1-)^2 K_0 > 0.
   
  We will use this below.
  
  Using Proposition <ref> we can find a μ-bubble Ω minimizing
  
      (Ω) = ∫_∂^*Ωλ  d^1 - ∫_Ω(χ_Ω-χ_Ω_0) h λ  d^2
  
  where Ω_0 is some reference Caccioppoli set B_(p) ⊂Ω_0 ⊂Σ∖ B_(q). By Lemma <ref>, k_γ = - λ^-1∇_Σλ,ν_γ + h. As in Lemma <ref>, we compute
  
      0     ≤∫_γ( |∇_γψ|^2 λ - K_Σψ^2 λ - k_γ^2 ψ^2 λ + (Δ_Σλ - Δ_γλ) ψ^2 .
         .            - ∇_Σλ,ν_γψ^2 h - ∇_Σ h ,ν_γψ^2 λ) d^1.
  
  Take ψ = λ^-1/2 to find
  
      0     ≤∫_γ( 1 4  λ^-2 |∇_γλ|^2  - K_Σ   - k_γ^2  +λ^-1 (Δ_Σλ - Δ_γλ)  .
         .           - λ^-1∇_Σλ,ν_γ h - ∇_Σ h ,ν_γ) d^1.
  
  Using 12 k_γ^2 =  12 λ^-2∇_Σλ,ν_γ^2 - λ^-1∇_Σλ,ν_γh +  12 h^2, we have
  
      0     ≤∫_γ( 1 4  λ^-2 |∇_γλ|^2  - K_Σ  +λ^-1 (Δ_Σλ - Δ_γλ)  .
         .          - 12 λ^-2∇_Σλ,ν_γ^2 -  12 h^2 - ∇_Σ h ,ν_γ) d^1
      
          = ∫_γ( -  3 4  λ^-2 |∇_γλ|^2  - K_Σ  +λ^-1 (Δ_Σλ)  .
         .          - 12 λ^-2∇_Σλ,ν_γ^2 -  12 h^2 - ∇_Σ h ,ν_γ) d^1,
  
  where we integrated by parts in the second step. Now, using the equation satisfied by λ, we have
  
      0     ≤∫_γ( - 3 4  λ^-2 |∇_γλ|^2  +  12 λ^-2 |∇_Σλ|^2 -   12 λ^-2∇_Σλ,ν_γ^2    .
         .         - K_0 -  12 h^2  - ∇_Σ h ,ν_γ) d^1.
  
  Note that |∇_Σλ|^2 = |∇_γλ|^2 + ∇_Σλ,ν_γ^2, so we find
  
      ∫_γ (K_0 +  12 h^2 + ∇_Σ h,ν_γ) d^1 ≤ 0
  
  However, we have seen above that K_0 +  12 h^2 + ∇_Σ h,ν_γ>0. This is a contradiction, completing the proof. 
  
  
  The following is the free-boundary analogue of the previous result. We note that Carlotto–Franz <cit.> have recently extended the original Schoen–Yau method <cit.> to the setting of stable free boundary minimal surfaces. As above, we use μ-bubbles to give an alternative proof of this fact.[In fact, we prove a slightly stronger statement than <cit.>, as needed for the warped product inductive descent argument from <cit.>.]
  
  
  For compact 2-dimensional Riemannian manifold (Σ^2,g) with boundary, suppose there is a smooth function λ > 0 so that 
  
      Δ_Σλ≤ - (K_0-K_Σ) λ +  12 λ^-1 |∇_Σλ|^2
  
  in Σ for some K_0 ∈ (0,∞). Suppose also that ∇_Σλ,η = - k_∂Σλ along ∂Σ for η the outwards pointing unit normal. Then, _g Σ≤√(2K_0)π. 
  
  
  If the diameter bound fails, we can argue precisely as in Lemma <ref> to find a free boundary μ-bubble Ω. Choose some curve γ in ∂Ω. If γ is a closed loop, the calculations in Lemma <ref> yield a contradiction. As such, we assume that γ : [a,b] →Σ is an arc with ∂γ⊂∂Σ. Using Proposition <ref>, the calculation above carries over to yield
  
      0     ≤∫_γ( 1 4  λ^-2 |∇_γλ|^2  - K_Σ  +λ^-1 (Δ_Σλ - Δ_γλ)  .
         .          - 12 λ^-2∇_Σλ,ν_γ^2 -  12 h^2 - ∇_Σ h ,ν_γ) d^1
              - (k_∂Σ(γ(b)) - k_∂Σ(γ(a))) .
  
  At this point, we integrate by parts and pick up boundary terms 
  
      0     ≤∫_γ( -  3 4  λ^-2 |∇_γλ|^2  - K_Σ  +λ^-1 (Δ_Σλ)  .
         .          - 12 λ^-2∇_Σλ,ν_γ^2 -  12 h^2 - ∇_Σ h ,ν_γ) d^1
              - ( λ(b)^-1∇_Σλ,η(γ(b)) -  λ(a)^-1∇_Σλ,η(γ(a)))
              - (k_∂Σ(γ(b)) - k_∂Σ(γ(a))) 
          = ∫_γ( -  3 4  λ^-2 |∇_γλ|^2  +  12 λ^-2 |∇_Σλ|^2   - 12 λ^-2∇_Σλ,ν_γ^2  .
         .           - K_0  -  12 h^2 - ∇_Σ h ,ν_γ) d^1.
  
  The proof is now completed as before. 
  
  
  
  
  
  § PROOF OF THEOREM <REF>
   
  
  For n=4,5, consider (N^n,g) a smooth closed aspherical n-manifold with a Riemannian metric of non-negative scalar curvature. Running the Ricci flow for a short time yields a metric with positive scalar curvature unless g is Ricci flat. However, if g is Ricci flat, the Cheeger–Gromoll splitting theorem <cit.> implies that the universal cover (Ñ,g̃) splits isometrically as (Ñ',g̃')×^k for Ñ' compact. Because N is aspherical, Ñ' is a point, so (Ñ,g̃) is flat ^n. 
  
  As such, it suffices to prove that N does not admit a metric of positive scalar curvature. We will assume that n=5, since if (N^4,g) is a compact aspherical 4-manifold with positive scalar curvature, then N × S^1 is aspherical (cf. <cit.>) and has positive scalar curvature when equipped with the product metric. 
  
  To summarize the above discussion, we can assume that (N^5,g) is a compact aspherical Riemannian 5-manifold with scalar curvature R_g ≥ 5. We write Ñ for the universal cover of N. By the results in Section <ref>, there is a geodesic line σ in the universal cover Ñ so that for any L>0 there is a compact two-sided hypersurface  with boundary M̂_4 in Ñ with d(∂M̂_4,σ()) ≥ 3L and so that M̂_4 has non-zero algebraic intersection with σ. We will take L sufficiently large below. 
  
  
  
   §.§ The σ-transversal 4-dimensional area-minimizer
   Find a smooth two-sided compact area-minimizing hypersurface M_4 homologous to M̂_4 relative to ∂ M_4 = ∂M̂_4. Stability of M_4 and R_g ≥ 5 implies that 
  
      ∫_M_4 (|∇_M_4ψ|^2 -  12 (5-R_M_4+ |A|^2)ψ^2   d^4 ≥ 0,    ∀ψ∈ C^1_0(M).
  
  As such, there is u_4 ∈ C^∞_0(M_4), u_4>0 on M_4, with 
  
      Δ_M_4 u_4 ≤ - 12 (5-R_M_4) u_4 .
  
  Our goal is to show that for L sufficiently large, we can find Ω_4 ⊂ M_4 so that ∂Ω_4 can be filled in by 4-chains that avoid σ(). Then, adding these chains to Ω_4 we find 4-cycle with non-trivial algebraic intersection with σ, contradicting H_4(Ñ) = 0. 
  
  
  
   §.§ The 3-dimensional μ-bubble
   Pick ρ_4 : Ñ→ a smoothing of d(·,σ()) so that |ρ_4| ≤ 2, ρ_4 ≥ L + 4π + 1 on ∂ M_4 and ρ_4 = 0 on M_4∩σ().  Fix >0 small so that L+4π + and L- are regular values of ρ_4 and then define 
  
      ρ̃_4(x) = ρ_4(x) - L - 2π/4 + 2/π,      h_4(x) = - tan(ρ̃_4(x))
  
  for x ∈ M_4' : = {ρ̃_4(x) ∈ (-π 2,π 2) }. Observe that 
  
      1 + h_4(x)^2 - 2 |∇ h_4| 
      ≥ 1 + tan^2(ρ̃_4(x)) - 2/4+2/π |ρ_4(x)| 1/cos^2(ρ̃_4(x))≥ 0
  
  on M_4'. Choosing a_4 ∼ 0 a regular value of ρ̃_4, we can set Ω_0 = ρ̃^-1((-∞,a_4)) and use Proposition <ref> to minimize
  
      _4(Ω) : = ∫_∂Ω u_4   d^3 - ∫_M_4 (χ_Ω - χ_Ω_0) h_4 u_4   d^4
  
  among Caccioppoli sets Ω with ΩΔΩ_0 ⋐ M_4'. Denote this minimizer by Ω_4 and let M_3: = ∂Ω_4. Note that M_3 is a cycle and 
  
      d(M_3,σ()) ≥ 12 (L-).
   
  
  
  Using (<ref>), (<ref>) and Lemma <ref> we see that
  
      ∫_M_3( |∇_M_3ψ|^2 u_4 -  12 (4-R_M_3) ψ^2 u_4 - (Δ_M_3 u_4) ψ^2 )   d^3 ≥ 0
  
  for all ψ∈ C^1(M_3). In particular, there is u_3 with
  
      _M_3 (u_4 ∇_M_3 u_3) ≤ -  12 (4-R_M_3) u_3 u_4 - (Δ_M_3 u_4)u_3.
  
  This implies that
  
      Δ_M_3 (u_3u_4)     = _M_3(u_3 ∇_M_3 u_4) +  _M_3(u_4 ∇_M_3 u_3) 
         ≤ -  12 (4-R_M_3) u_3 u_4 + ∇_M_3 u_3,∇_M_3 u_4.
  
  As mentioned above, our goal is now to show that each component of M_3 can be filled by a chain with uniformly bounded diameter as L→∞. Note that M_3 is Yamabe positive but it might hold that the diameter of each component of M_3 is unbounded as L→∞. 
  
  
  
   §.§ A slice and dice procedure for M_3
   We can consider each component of M_3 separately, so it suffices to assume that M_3 is connected. It will be important later to observe that M_3 is orientable, by construction. 
  
  Set λ_3 : = u_3u_4>0. For Σ^2 ⊂ M_3, we define 
  
      _3(Σ) = ∫_Σλ_3 .
  
  Arguing as in <cit.>, we can derive the following result for stable critical points of _3. 
  
  Suppose that Σ is a connected (two-sided) stable critical point of _3. Then Σ is a topological sphere with ^2(Σ) ≤ 2π and Σ≤π. 
  
  
  Lemma <ref> implies that H = - λ_3^-1∇_M_3λ_3,ν. Similarly, Lemma <ref> combined with (<ref>) yields 
  
      0    ≤∫_Σ(|∇_Σψ|^2 λ_3 - (2   - K_Σ )ψ^2 λ_3 - (Δ_Σλ_3)ψ^2    .
         .          + ∇_M_3u_3,∇_M_3u_4ψ^2  - 12 λ_3^-1∇_M_3λ_3,ν^2ψ^2   
       )d^2.
  
  Note that 
  
      2 λ_3 ∇_M_3u_3,∇_M_3u_4 -  ∇_M_3λ_3,ν^2   ≤ |∇_M_3λ_3|^2  - ∇_M_3λ_3,ν^2  = |∇_Σλ_3|^2
  
  This yields
  
      0 ≤∫_Σ(|∇_Σψ|^2 λ_3 - (2   - K_Σ )ψ^2 λ_3 - (Δ_Σλ_3)ψ^2 +  12 λ_3^-1  |∇_Σλ_3|^2   ψ^2   
       )d^2
  
  Take ψ = λ_3^-1/2 to find
  
      0    ≤∫_Σ( 34 λ_3^-2 |∇_Σλ_3|^2  - (2   - K_Σ )  - λ_3^-1(Δ_Σλ_3)  )d^2.
  
  Integrating by parts on the last term, this implies
  
      2 ^2(Σ) ≤∫_Σ K_Σ  d^2 = 2πχ(Σ).
  
  As such, Σ is a topological sphere and ^2(Σ) ≤ 2π. 
  
  Returning to (<ref>), we find a smooth function u_2>0 on Σ with 
  
      _Σ(λ_3 ∇_Σ u_2) ≤ - (2-K_Σ) u_2λ_3 - (Δ_Σλ_3)u_2  +  12 λ_3^-1  |∇_Σλ_3|^2    u_2.
  
  Set λ_2 = u_2λ_3 = u_2u_3u_4. We compute
  
      Δ_Σλ_2  ≤ - (2-K_Σ) λ_2 + ∇_Σ u_2,∇_Σλ_3  +  12 λ_3^-1  |∇_Σλ_3|^2 .
  
  Now, we observe that 
  
      λ_2^-1 |∇_Σλ_2|^2 ≥ 2 ∇_Σ u_2,∇_Σλ_3 + λ_3^-1 |∇_Σλ_3|^2,
  
  so 
  
      Δ_Σλ_2  ≤ - (2-K_Σ) λ_2 +  12 λ_2^-1 |∇_Σλ_2|^2 .
  
  As such, the diameter bounds follow from Lemma <ref>. 
  
  
  We first use _3-minimization to slice M_3 into a manifold with simple second homology. This will allow us to use free boundary μ-bubbles to dice the resulting manifold into pieces that can be filled within bounded distance. 
  
  There are Σ̂_1,…,Σ̂_k⊂ M_3 pairwise disjoint two-sided stable critical points of _3 so that the manifold with boundary M̂_3 := M_3 ∖ (∪_j=1^k Σ̂_j) is connected and has H_2(∂M̂_3) → H_2(M̂_3) surjective. 
  
  
  We proceed inductively. Assume that for k≥ 0 we have chosen 
  
      Σ̂_1,…,Σ̂_k ⊂ M_3
  
  pairwise disjoint two-sided stable critical points of _3. Suppose that for M̂_3^(k) := M_3 ∖ (∪_j=1^k Σ̂_j), the map H_2(∂M̂_3^(k)) → H_2(M̂_3^(k)) is not surjective. Consider α not in the image. Minimize _3 area in the homology class. Because each component of ∂M̂_3^(k) is smooth and stationary for _3, we can find a representative of α 
  
      α = [Σ'_1] + … + [Σ'_ℓ]
  
  with the connected surfaces Σ_1',…,Σ_ℓ' pairwise disjoint two-sided stable critical points. Moreover, each Σ_j' is either contained in the interior of M̂_3^(k) or coincides with a component of ∂M̂_3^(k). By choice of α, there must be some j with Σ_j' in the interior of M̂_3^(k), not separating, and not in the image of H_2(∂M̂_3^(k)). Define Σ̂_k+1 = Σ'_j. 
  
  It suffices to show that this process terminates. If not, we obtain a sequence Σ̂_1,…,Σ̂_k,… of stable critical points of _3. By Lemma <ref>, each Σ̂_k has bounded area. Moreover, by a standard blow-up argument using <cit.>, the Σ̂_k have uniform curvature bounds. As such, passing to a subsequence, the Σ̂_k_1,Σ̂_k_2,… are converging as one-sheeted graphs to some Σ̂_∞. However, this clearly contradicts the construction of the Σ̂_j above, since it shows that Σ̂_k_j+1 is homologous to Σ̂_k_j in M̂_3^(k_j+1), for j large. This completes the proof.
  
  
  Choose M̂_3 as in the previous lemma. We recall that for Ω⊂M̂_3, we write ∂Ω for the topological boundary. Suppose that Ω⊂M̂_3 is a connected region with corners and Ω≠ M. We assume that ∂Ω consists of smooth properly embedded surfaces in M̂_3.
  
  A connected component of M̂_3 ∖Ω contains exactly one component of ∂Ω. 
  
  
  Clearly a component of M̂_3 ∖Ω contains at least one component of ∂Ω. Suppose, for contradiction, that some component Ω' of M̂_3 ∖Ω contained two components of ∂'Ω, Σ_1 and Σ_2. For j=1,2, take a point p_j∈Σ_j. Then we can find curves γ∈Ω, γ'∈Ω' connecting p_1, p_2. The concatenated curve σ=γγ' is then an embedded S^1 that intersects Σ_1 transversely in precisely one point. This implies that [σ] is not torsion in H_1(M̂_3). 
  
  On the other hand, the long exact sequence in homology for a pair yields
  
      H_2(∂M̂_3) →  H_2(M̂_3)→ H_2(M̂_3,∂M̂_3) → H_1(∂M̂_3) = 0.
  
  The final term vanishes since ∂M̂_3 consists of spheres, by Lemma <ref>. Thus we conclude that H_2(M̂_3,∂ M_3) = 0. Lefschetz duality (cf. <cit.>) implies that H^1(M̂_3) = 0, so the universal coefficients theorem implies that H_1(M̂_3) is torsion. This is a contradiction, finishing the proof. 
  
  
  We now show how to inductively dice M̂_3 by finitely many (guaranteed by the condition (1) below, since M̂_3 is compact) free boundary μ-bubbles of controlled diameter and boundary behavior. Fix p in the interior of M̂_3. For technical reasons, we start by choosing Ω_1 = B_(p), chosen so that B_(p) is contained in the interior of M̂_3. Assume that we have chosen regions 
  
      Ω_1 ⊂Ω_2 ⊂…⊂Ω_k
  
  with the following properties:
  
  
    * d_M̂_3(∂Ω_j+1,∂Ω_j) ≥2π3,
  
    * each component of Ω_j+1∖Ω_j has diameter at most 10π
  
    * any component Υ⊂∂Ω_j has Υ≤π, and
  
    * each component of ∂Ω_j is either a topological sphere or a topological disk with boundary in ∂M̂_3.
  
  
  Assume that Ω_k ≠M̂_3 and that there is some p ∈M̂_3 with d(p,Ω_k) > 4π (otherwise we set Ω_k+1 = M̂_3). 
  
  We now choose a μ-bubble based on a smoothing ρ of d(·,Ω_k) such that d(·, Ω_k)≤ρ≤ 32 d(·, Ω_k), |ρ| <  32 and ρ|_Ω_k = 0 and then taking
  
      h(x) = -tan (  1 3 (ρ(x)-π) - π 2 ).
  
  Observe that 
  
      1 + h(x)^2 - 2|∇ h| ≥ 0,
  
  by a similar calculation to the one used above. 
  
  By Proposition <ref>, we can find a free boundary μ-bubble Ω_k+1 inside {π≤ρ≤ 4π} minimizing 
  
      _3(Ω) = ∫_∂^*Ωλ_3   d^2 - ∫_Ω (χ_Ω - χ_Ω_0) λ_3 h   d^3.
   
  Choose a component Σ of ∂Ω_k+1. Clearly condition (1) is satisfied since h→∞ at the surface {ρ=π}, which satisfies d(·,Ω_k)≥2π3.
  
  
  
  Ω_k+1 satisfies property (2). 
  
  
  By Lemma <ref>, if p,q are in the same connected component of Ω_k+1∖Ω_k, then they are connected there to a unique component of ∂Ω_k. Call this component Σ_k. By construction d(p,Σ_k),d(q,Σ_k) ≤ 4π. Moreover, Σ_k ≤π. Thus, d(p,q) ≤ 9π. This completes the proof. 
  
  
  
  We now verify that Ω_k+1 satisfies conditions (3) and (4). Take a component Σ of ∂Ω_k+1. By Proposition <ref>, Σ has (possibly empty) free boundary at ∂M̂_3. If Σ has no boundary, then Σ satisfies (<ref>), and thus by the same proof as in Lemma <ref>, Σ is a topological sphere with Σ≤π. If Σ does have boundary, we can conclude that 
  
      0    ≤∫_Σ(|∇_Σψ|^2 λ_3 -12 (R_M̂_3 - 1 - 2K_Σ  )ψ^2 λ_3 + (Δ_M̂_3λ_3  -Δ_Σλ_3)ψ^2     .
         .          - 12 λ_3^-1∇_M̂_3λ_3,ν_Σ^2ψ^2  
      )d^2
              - ∫_∂Σ A_∂M̂_3 (ν_Σ,ν_Σ) ψ^2 λ_3   d^1 .
  
  Using (<ref>) (recall that λ_3 = u_3u_4), we find
  
      0    ≤∫_Σ(|∇_Σψ|^2 λ_3 -  (1 - K_Σ  )ψ^2 λ_3 - (Δ_Σλ_3)ψ^2     .
         .          + ∇_M̂_3u_3,∇_M̂_3u_4ψ^2 - 12 λ_3^-1∇_M̂_3λ_3,ν_Σ^2ψ^2  
      )d^2
              - ∫_∂Σ A_∂M̂_3 (ν_Σ,ν_Σ) ψ^2 λ_3   d^1 .
  
  Taking first ψ = λ_3^-1/2, we conclude that 
  
      0    ≤∫_Σ( 1 4 λ_3^-2 |∇_Σλ_3|^2  -  (1 - K_Σ  )  - λ_3^-1 (Δ_Σλ_3)     .
         .          + λ_3^-1∇_M̂_3u_3,∇_M̂_3u_4 - 12 λ_3^-2∇_M̂_3λ_3,ν_Σ^2 
      )d^2
              - ∫_∂Σ A_∂M̂_3 (ν_Σ,ν_Σ)     d^1
          = ∫_Σ( K_Σ - 1  - 3 4 λ_3^-2 |∇_Σλ_3|^2 + λ_3^-1∇_M̂_3u_3,∇_M̂_3u_4.
                 . - 12 λ_3^-2∇_M̂_3λ_3,ν_Σ^2 
      )d^2
              - ∫_∂Σ (λ_3^-1∇_Σλ_3,η + A_∂M̂_3 (ν_Σ,ν_Σ)  )   d^1
          = ∫_Σ( K_Σ - 1  - 3 4 λ_3^-2 |∇_Σλ_3|^2 + λ_3^-1∇_M̂_3u_3,∇_M̂_3u_4.
         .          - 12 λ_3^-2∇_M̂_3λ_3,ν_Σ^2 
      )d^2
              - ∫_∂Σ( - λ_3^-1∇_Σλ_3,ν_∂M̂_3 + H_∂M̂_3 - A_∂M̂_3 (τ_∂Σ,τ_∂Σ) )   d^1
          = ∫_Σ( K_Σ - 1  - 3 4 λ_3^-2 |∇_Σλ_3|^2 + λ_3^-1∇_M̂_3u_3,∇_M̂_3u_4.
                 .- 12 λ_3^-2∇_M̂_3λ_3,ν_Σ^2 
      )d^2
              + ∫_∂Σ A_∂M̂_3 (τ_∂Σ,τ_∂Σ)    d^1.
  
  Here η is the outward conormal vector of ∂Σ⊂Σ, τ_∂Σ is the unit tangential vector along ∂Σ. Since Σ meets ∂M̂_3 orthogonally, A_∂M̂_3(τ_∂Σ,τ_∂Σ)=k_∂Σ. Also, as in Lemma <ref>, the terms involving λ_3 are non-positive. Therefore we find 
  
      ^2(Σ) ≤∫_Σ K_Σ  + ∫_∂Σ k_∂Σ  d^1 = 2πχ(Σ).
  
  This implies that Σ is a topological disk.
  
  Finally, as in Lemma <ref> we find u_2 satisfying 
  
      _Σ(λ_3 ∇_Σ u_2) ≤ - (2-K_Σ) u_2λ_3 - (Δ_Σλ_3)u_2  +  12 λ_3^-1  |∇_Σλ_3|^2 u_2    in Σ
      ∇_Σ u_2,η = A_∂M̂_3(ν_Σ,ν_Σ)u_2    on ∂Σ
  
  Thus, we have that λ_2=u_2λ_3 =u_2u_3u_4 satisfies
  
      Δ_Σλ_2  ≤ - (2-K_Σ) λ_2 +  12 λ_2^-1 |∇_Σλ_2|^2 .
  
  in Σ and 
  
      ∇_Σλ_2,η    =  A_∂M̂_3(ν_Σ,ν_Σ)  λ_2 + ∇_Σλ_3,η u_2 
          =   H_∂M̂_3λ_2  - λ_3^-1∇_Σλ_3,ν_∂M̂_3λ_2   -  k_∂Σλ_2 
          =  -  k_∂Σλ_2.
  
  Thus, Lemma <ref> implies that Σ≤π.
  
  
  
   §.§ Filling M_3
   We now show that M_3 can be filled in Ñ by a 4-chain of uniformly bounded diameter (as L→∞). As explained above, this will give the desired contradiction, proving Theorem <ref>. We first summarize the slice and dice procedure used above as it applies to M_3.
  
  There exists a set of disjoint embedded spheres Σ̂_1,…,Σ̂_k ⊂ M_3 with Σ̂_j ≤π. Moreover, there is a set Υ̂_1,…,Υ̂_ℓ⊂ M of embedded disks with Υ̂_j≤π and ∂Υ̂_j contained in ∪_i=1^k Σ̂_i and so that the interiors of Υ̂_j are pairwise disjoint with each other as well as with each Σ̂_j. Finally, each connected component U_1,U_2,…,U_m  of 
  
      M_3 ∖ ((∪_j=1^k Σ̂_j) ∪ (∪_j=1^ℓΥ̂_j))
  
  has diameter bounded by 10π. 
  
  Observe that any component of ∂ U_j is a topological sphere (it will be smooth except it could have a right angle corner along one or two closed curves arising from points where the free boundary disks intersect the original spheres) of extrinsic diameter at most 3π. Write these spheres as Γ_j^1,…,Γ_j^n(j). By Proposition <ref>, there is R>0 (independent of L) so that we can fill the Γ_j^i by Γ̂_j^i with extrinsic diameter at most R. Then, 
  
      U_j - Γ̂_j^1 - … - Γ̂_j^n(j)
  
  is a cycle with extrinsic diameter at most 2R+10π. As such, there is R̂>0 (independent of L) so that by Proposition <ref> we can find a 4-chain Û_j of extrinsic diameter at most R̂ with
  
      ∂Û_j = U_j - Γ̂_j^1 - … - Γ̂_j^n(j) .
  
  Note that 
  
      M_3 - ∑_j=1^m ∂Û_j =  ∑_j=1^m ∑_i=1^n(j)Γ̂_j^i.
  
  Note that for each Γ_j^i, there is an index u(j,i) ∈{1,…,k} so that Γ_j^i intersects the sphere Σ̂_u(j,i) but not any of the other spheres. As such, we group the Γ_j^i by u(j,i) = 1,2,…,k. We find that for a ∈{1,2,…,k},
  
      ∑_{i,j : u(i,j) = a}Γ̂_j^i
  
  is a cycle of diameter at most 2R + π. Indeed, 
  
      ∂[ ∑_a=1^k ∑_{i,j : u(i,j) = a}Γ̂_j^i ]= 0
  
  and if 
  
      ∂[ ∑_{i,j : u(i,j) = a}Γ̂_j^i ] ≠ 0
  
  for some a ∈{1,…,k}, then this boundary component intersects there sphere Σ̂_a (but none of the other spheres). As such, the other terms in the sum (<ref>) could not cancel this term. This would be a contradiction.
  
  
  
  As such, there is R̃>0 (independent of L) so that by Proposition <ref> yet again, there is a 4-chain Ξ_a with 
  
      ∂Ξ_a = ∑_i,j : u(i,j) = aΓ̂_j^i
  
  and extrinsic diameter at most R̃. Thus, we have written 
  
      M_3 = ∂[ ∑_j=1^m Û_j + ∑_a=1^k Ξ_a ]
  
  where each term in the sum has diameter uniformly bounded as L→∞. This completes the proof.
  
  
  
  
  
   §.§ The Uryson width of compact conformally PSC 3-manifold
  
  As a simple consequence of the slice–dice argument in Section <ref>, we obtain the following structure theorem for compact 3-manifold with uniformly positive scalar curvature.
  
  
  	Let (M,g) be a closed Riemannian manifold. Suppose there exists u>0 on M such that R(u^4 g)>3. Then M admits a continuous map φ to a graph G such that _g(φ^-1(p))<10 π for any p∈ G. 
  
  
  
  	By definition, this implies that first Uryson width (see, e.g. <cit.>)of M is bounded by 10π. Previously, this has been proved when π_1(M) is trivial in <cit.> and when π_1(M) is finite or π_1(M)= in <cit.>. See also <cit.>.
  
  
  
  	We first treat the case where H^1(M)=0. In this case, as in Section <ref>, we can find an exhaustion of M by regions
  	
      Ω_1⊂⋯Ω_k
  
  	such that properties (1)–(4) holds. We then construct a graph as follows. Let v_1 be a vertex corresponding to Ω_1. Suppose we have constructed a 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § PROOF OF THEOREM <REF>
   
  
  Let n≥ 3. For X a n-manifold (compact or non-compact), suppose that g is a complete metric on M = T^n# X with non-negative scalar curvature. By a result of Kazdan <cit.>, either g is Ricci flat or M admits a complete metric of positive scalar curvature. However, a complete Ricci flat metric on T^n# X is easily seen to be flat (this follows from the splitting theorem; for example, see <cit.>). As such, it suffices to consider the case of positive scalar curvature.  We will pass to an appropriate covering space and apply the μ-bubble technique on the cover, after carefully choosing a weight function h. 
  
  Fix >0 small and define 
  
      Ξ : =  {x⃗= (x_1,⋯,x_n) ∈^n : |x⃗ - k⃗| > , k⃗∈^n}/∼
  
  where (x_1,⋯,x_n) ∼ (x_1+k_1,⋯,x_n+k_n) for k_1,⋯,k_n ∈. By assumption, there is a map Ψ : Ξ→  M so that Ψ is a diffeomorphism onto its image. By scaling, we can assume that R_g > 1 on Ψ(Ξ).
  
  
  Observe that M is (topologically) covered by M̂ = (T^n-1×) #_ X (unwind one of the S^1 factors in T^n). Define
  
      Ξ̂= {x⃗=(x_1,⋯,x_n) ∈^n : |x⃗ - k⃗| > , k⃗∈^n}/∼
  
  where (x_1,⋯,x_n) ∼ (x_1+k_1,⋯, x_n-1+k_n-1,x_n) and note that the map Ψ lifts to Ψ̂:Ξ̂→X̂, a diffeomorphism onto its image M̂_0. It is useful to write 
  
      M̂ = M̂_0 ∪(∪_k∈ X_k)
  
  where each X_k is (topologically X∖  B for an n-balls B in X) attached to M̂_0 : = Ψ̂(Ξ̂) along small spheres centered at (0,0,k). 
  
  We now define a function ρ_0 : M̂→ as follows. Define Ξ̂_2 as in (<ref>) but with 2 in the place of . On Ψ(Ξ̂_2), we take ρ_0(x_1,⋯,x_n) = x_n. On the annuli Ψ̂(Ξ̂_2∖Ξ̂) centered at (0,⋯,0,k), interpolate between x_n and k+ 12 (we can do this with uniformly C^1-norm independent of k). Then, on X_k define 
  
      ρ_0(p) = k +  12 + _g(p,∂ X_k)   k≥ 0;
       k+12 - _g(p,∂ X_k)   k<0.
  
  We now define ρ_1 to be a smoothing of ρ_0. We can assume that ρ_1≡ k+ 12 in a small neighborhood of ∂ X_k. Since M̂ is a covering space of M, there is L>0 so that 
  
      |(ρ_1)|_g < L.
  
  We may take L larger if necessary to assume that π L2 = J +  34 for some J ∈. 
  
  We now define a function h∈ C(M̂,[-∞,∞]) as follows. On M̂_0 ∩{ -π L2≤ρ_1 ≤π L2}, we define 
  
      h(p) = - tan(1Lρ_1(p)).
  
  On the rest of M̂_0 we set h = ±∞ such that it is continuous to [-∞,∞]. We now define h on X_k. When k>J, set h=-∞ on X_k, and when K<-J, set h=∞ on X_k. Now assume |k| ≤ J.
  
  
  For 0≤ k≤ J and
  
      p ∈ X_k ∩{ρ_1 < k +  12 + 2L/tan(L^-1(k+ 12))},
  
  or -J≤ k<0 and
  
      p ∈ X_k ∩{ρ_1 > k +  12 + 2L/tan(L^-1(k+ 12))},
  
   we set
  
      h(p) =  2L/ρ_1(p) - (k+ 12 ) -  2L/tan ( L^-1(k+ 12)) .
  
  Otherwise we set h(p)=±∞ such that h is continuous.
  
  
   
  We make several observations. First of all, since -J ≤ k≤ J, we see that 
  
      -π 2 < L^-1(k+ 12) < π 2.
  
  Moreover, for p ∈∂ X_k, we have that
  
      h(p) = - tan (L^-1(k+ 12)) = - tan(L^-1ρ_1(p)),
  
  and thus h is Lipchitz across ∂ X_k. Finally, if 0≤ k ≤ J, p∈X_k and 
  
      ρ_1(p) ↗ k +  12 + 2L/tan(L^-1(k+ 12)),
  
  we have that h(p) → -∞. Similarly, if -J≤ k ≤ 0, p∈X_k and
  
      ρ_1(p) ↘ k +  12 + 2L/tan(L^-1(k+ 12)),
  
  h_1(p)→∞. Thus h is continuous.
  
  Note that {|h| < ∞} is compact. This is because this region is compact in M_0, only finitely many ends X_k are included in this set, and in each X_k, the region where {|h|<∞} is bounded. 
  
   We can smooth h slightly to find a function h∈ C^∞(M̂) satisfying
  
      R_ĝ + h^2 - 2|∇ h| > 0
  
  on {|h|<∞}. 
  
  
  
  The function h constructed above is smooth away from ∂ X_k (and Lipschitz there). As such, if we prove (<ref>) for function h considered above, then we can easily find a smooth function satisfying (<ref>). 
  
  
  We first check (<ref>) on M̂_0. There, R_ĝ > 1. As such, we have that
  
      R_ĝ + h^2 - 2|∇ h| > 1 + tan^2(L^-1ρ_1(p)) - 2 cos^-2(L^-1ρ_1(p)) > 0.
  
  On the other hand, on X_k (we assume that k≥ 0 as the k<0 case is similar), we only know that R_ĝ > 0. Nevertheless, we compute
  
      R_ĝ + h^2 - 2|∇ h| >  4L^2 - 4L^2/( ρ_1(p) - (k+ 12 ) -  2L/tan ( L^-1(k+ 12)))^2 = 0.
  
  This completes the proof. 
  
  We can thus consider μ-bubbles with respect to the function h we have just defined. We fix 
  
      Ω_0 : =(Ψ̂(Ξ̂∩{x_n < - 12}) ∪ (∪_k<0 X_k))∩{|h|<∞}.
  
  We can minimize  among all Cacioppoli sets Ω such that ΩΔΩ_0 is compactly contained in {|h|<∞} by the argument given in Proposition <ref> (with u=1). Denote by Ω the connected component of the minimizer containing {ρ_1=-J}. Since n≤ 7, each component of ∂Ω is compact and regular. By the stability inequality for  from Lemma <ref> (with u=1) and (<ref>), we see that Σ = ∂Ω satisfies 
  
      ∫_Σ( |∇φ|^2 +  12  R_Σφ^2 ) d^n-1 > 0
  
  for all φ∈ C^∞(Σ). 
  
  We can find a compact region M̂' ⊂M̂ with smooth boundary so that ∂Ω⊂M̂'. Furthermore, we can arrange that ∂M̂' ∩M̂_0 = Ψ̂({z = ± (J+1)}). Note that the other boundary components of M̂' thus lie completely in some X_k. 
  
  In particular, 
  ∂M̂∖M̂_0 bounds some compact manifold with boundary. Cap these components off and then glue the {z=J+1} and {z=-(J+1)} tori to each other. We thus find a manifold M̃ diffeomorphic to T^n #X̃ for X̃ closed and Σ^n-1⊂M̃ a hypersurface that is homologous to [T^n-1×{*}]∈ H_n-1(M̃) that satisfies (<ref>). (We have not constructed a metric on M̃, but this does not matter in the remaining part of the argument, all we need is the topology of M̃ and Σ as well as the fact that Σ satisfies (<ref>).) 
  
  We claim that this leads to a contradiction following the argument in <cit.>. Indeed, on the one hand (<ref>) implies that (each component of) Σ has positive first eigenvalue of the conformal Laplacian, and thus admits a metric of positive scalar curvature. On the other hand, we can pull back the 1-forms dx^1,…,dx^n-1 along the map π: (T^n#X̃) → T^n to find 1-forms ω^1,…,ω^n-1 so that 
  
      ∫_Σω^1 ∧…∧ω^n-1≠ 0.
  
  (this follows from the fact that Σ is homologous to T^n-1×{*}). The proof can now be completed using the inductive method of <cit.>. 
  
  
  
  
  § SCHOEN-YAU-SCHICK MANIFOLDS
  
  In this section, we briefly indicate an extension of Theorem <ref> to manifolds in the form of (M× S^1)#X, where M is a Schoen–Yau–Schick manifold (abbreviated as SYS manifold, following the definition of Gromov <cit.>). These manifolds was first considered in the celebrated work <cit.> of Schoen-Yau where the inductive descent argument was introduced. In <cit.>, these manifolds are said to be of class C_n. We recall the definition here.
  
  [<cit.><cit.><cit.><cit.>]
  	Let n≥ 2. A compact orientable n-manifold M is called Schoen–Yau–Schick, if there exist n-2 integer homology classes h_1,⋯,h_n-2∈ H^1(M) such that σ= h_1⌢⋯⌢ h_n-2⌢ [M]∈ H_2(M,) is non-spherical. That is, σ is not contained in the image of the Hurewicz homomorphism π_2(M)→ H_2(M).
  
  
  For example, the torus is an SYS manifold. Using minimal surface and induction descent argument, Schoen–Yau in <cit.> proved that SYS manifolds of dimension at most 7 does not admit positive scalar curvature metrics. In <cit.>, Schick constructed an SYS manifold which is a counterexample to the unstable Gromov-Lawson-Rosenberg conjecture. 
  
  Let n≤ 6, and M^n be an SYS manifold. By passing to the cover M× and constructing the same functions ρ_0,ρ_1 and h as in Section 6, we can extend Theorem <ref> to the following.
  
  
  	Let 2≤ n≤ 6, M^n be an Schoen–Yau–Schick manifold. For any (n+1)-manifold X, the connected sum (M× S^1)#X does not admit a complete metric of positive scalar curvature.
  
  
  
  
  After the first version of this paper first appeared, S. Chen <cit.> generalized the proof of Theorem <ref> to show that there is no complete positive scalar curvature metric on M# X for M^n a Schoen–Yau–Schick manifold, 3 ≤ n≤ 7 and X^n an arbitrary manifold. 
  
  
  
  
  
  § THE LESOURD–UNGER–YAU REDUCTION
  
  
  Schoen–Yau demonstrated that the Liouville theorem stated here in Corollary <ref> is equivalent to a certain positive (non-negative) mass theorem for asymptotically flat manifolds with other ends <cit.> (cf. <cit.>) The usual positive mass theorem can be reduced to the non-existence of a positive scalar curvature metric on T^n # X (for X compact) by an argument of Lohkamp. However, in general this argument involves global conformal deformations as first developed by Schoen–Yau in their proof of the positive mass theorem <cit.>. 
  
  As such, it is not immediate that in the context of Corollary <ref>, it suffices to prove Theorem <ref>. This has been verified by Lesourd–Unger–Yau <cit.> (cf. <cit.>). For completeness, we indicate a proof of their reduction here.
  
  
  By work of Schoen–Yau <cit.>, to prove Corollary <ref>, it suffices to consider 3≤ n≤ 6 and a complete Riemannian n-manifold (M^n,g) with R_g≥ 0 and a conformal map Φ : M→ S^n. Denote by g_0 the round metric on S^n and write Φ^* g_0 = |Φ'|^2 g. 
  
  By <cit.>, the conformal Laplacian
  
      L_g = -Δ_g + n-2/4(n-1) R_g
  
  admits a minimal Greens function G>0 satisfying L_gG = δ_x_0 for every x_0 ∈ M. Denote by G_0 the Green's function for L_0 (the conformal Laplacian on (S^n,g_0)) based at Φ(x_0) = y_0. It is a standard fact that 
  
      G_0^4/n-2 g_0 = π^*(g_^n),
  
  where π : S^n ∖{y_0}→^n is the stereographic projection map. Pull this equation back by Φ to and define 
  
      g : = (G_0 ∘Φ)^4/n-2 |Φ'|^2 g = (Φ∘π)^* g_^n
  
  on M∖Φ^-1(y_0). Thus, g is flat (but likely incomplete). Observe, however that x_0 has a neighborhood U so that g|_U∖{x_0} is isometric to g_^n |_B_R(0)^c via the map Φ∘π. 
  
  Motivated by this fact, we define
  
      G = |Φ'|^n-2/2 G_0 ∘Φ
  
  and check that (see <cit.>)
  
      L_gG = ∑_P ∈Φ^-1(y_0) |Φ'(P)|^n+2/2δ_P.
  
  By construction (cf. <cit.>) we have that |Φ'(x_0)|^n+2/2 G ≤G. 
  
  We now define g_ = (G+)^4/n-2 g. Note that for >0, this is complete and (cf. <cit.>)
  
      R_g_ = 4(n-1)/n-2 (G+)^-n+2/n-2L_(G+) =    (G+)^-n+2/n-2 R_g ≥ 0.
  
  Moreover, g_ = ((G+)/G)^4/n-2g = (Φ∘π)^*(v_^4/n-2 g_^n), where 
  
      v_∘Φ∘π = G/G  + /G : = (v + ṽ) ∘Φ∘π.
  
  Observe that v is harmonic on B_R(0)^c⊂^n and limits to |Φ'(x_0)|^n+2/2 as x→∞. This follows from the fact that 
  
      G, |Φ'(x_0)|^-n+2/2G_0 = c_nr^n-2(1+o(1))
  
  near x_0, for some dimensional constant c_n > 0. This is easily proven by a scaling argument (cf. <cit.>, but note that the full power of that result is not needed here).
  
  Moreover, ṽ(x) = c_n^-1 |Φ'(x_0)|^n+2/2 |x|^2-n + O(|x|^1-n) as x→∞. Thus, we find that v_ can be expanded as
  
      v_(x) = A + B_ |x|^2-n + O(|x|^1-n)
  
  for some A >0. By a well-known argument due to Lohkamp (see <cit.>), if B_ < 0, we can modify g_ only in the chart (Φ∘π)(B_R(0)^c) and obtain a complete metric g_' on M∖Φ^-1(y_0) with g_' flat on (Φ∘π)(B_2R(0)^c) and so that g_' has non-negative scalar curvature. Quotienting the sides of a large box in the flat region thus yields a contradiction to Theorem <ref>. 
  
  Thus, we find that B_≥ 0. Sending → 0, we thus find a harmonic function v on B_R(0)^c ⊂^n with
  
      v(x) = A + B |x|^2-n + O(|x|^1-n)
  
  for some A>0,B≥ 0. Observe that v(x) ≤ A on B_R(0)^c. This follows from the construction of v and the fact that G was a minimal Greens function as discussed above. We can now apply the maximum principle to conclude that
  
      - B|x|^2-n + O(|x|^1-n) = A-v(x) ≥(min_∂ B_2R(0) (A-v))|x/(2R)|^2-n
  
  Because B ≥ 0, we thus see that v(x) ≡ A. This implies that G=G, completing the proof. 
  
  
  amsplain
  
  