Sewing Riemannian Manifolds with Positive Scalar Curvature
SSEWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALARCURVATURE
J. BASILIO, J. DODZIUK, AND C. SORMANIA bstract . We explore to what extent one may hope to preserve geometric prop-erties of three dimensional manifolds with lower scalar curvature bounds underGromov-Hausdor ff and Intrinsic Flat limits. We introduce a new construction,called sewing, of three dimensional manifolds that preserves positive scalar cur-vature. We then use sewing to produce sequences of such manifolds which con-verge to spaces that fail to have nonnegative scalar curvature in a standard gen-eralized sense. Since the notion of nonnegative scalar curvature is not strongenough to persist alone, we propose that one pair a lower scalar curvature boundwith a lower bound on the area of a closed minimal surface when taking se-quences as this will exclude the possibility of sewing of manifolds.
1. I ntroduction
In this paper we study three dimensional manifolds with positive scalar curva-ture. The scalar curvature of a Riemannian manifold is the average of the Riccicurvatures which in turn is the average of the sectional curvatures. It can be deter-mined more simply by taking the following limit:(1) Scal( p ) = lim r →
30 Vol E ( B (0 , r )) − Vol M ( B ( p , r )) r Vol E ( B (0 , r ))where Vol E ( B (0 , r )) = (4 / π r and Vol M ( B ( p , r )) is the Hausdor ff measure ofthe ball about p of radius r in our manifold, M .In [Gro14b], Gromov asks the following pair of deliberately vague questionswhich we paraphrase here: Given a class of Riemannian manifolds, B , what isthe weakest notion of convergence such that a sequence of manifolds, M j ∈ B ,subconverges to a limit M ∞ ∈ B where now we will expand B to include singu-lar metric spaces? What is this generalized class of singular metrics spaces thatshould be included in B ? Gromov points out that when B is the class of Rie-mannian manifolds with nonnegative sectional curvature then the “best known”answer to this question is Gromov-Hausdor ff convergence and the singular limitspaces are then Alexandrov spaces with nonnegative Alexandrov curvature. When B is the class of Riemannian manifolds with nonnegative Ricci curvature, one usesGromov-Hausdor ff and metric measure convergence to obtain limits which aremetric measure spaces with generalized nonnegative Ricci curvature as in work of J. Basilio was partially supported as a doctoral student by NSF DMS 1006059.C. Sormani was partially supported by NSF DMS 1006059. a r X i v : . [ m a t h . DG ] N ov J. BASILIO, J. DODZIUK, AND C. SORMANI
Cheeger-Colding [CC97]. Work towards defining classes of singular metric mea-sure spaces with generalized notions of nonnegative Ricci has been completed byLott-Villani, Sturm, Ambrosio-Gigli-Savare and others [LV09] [Stu06a] [AGS14].Gromov then writes that “the most tantalizing relation B is expressed with thescalar curvature by Scal ≥ k” [Gro14b]. Bamler [Bam16] and Gromov [Gro14a]have proven that under C convergence to smooth Riemannian limits Scal ≥ ≥ ff and metric measure limits [SW10]. Intrinsic flat conver-gence is a weaker notion of convergence in the sense that there are sequences ofmanifolds with no Gromov-Hausdor ff limit that have intrinsic flat limits, includ-ing Ilmanen’s Example of a sequence of three spheres with positive scalar curva-ture [SW11]. The third author has investigated intrinsic flat limits of manifoldswith nonnegative scalar curvature under additional conditions with Lee, Huang,LeFloch and Stavrov [LS14][HLS17][LS15] [SS17]. These papers support Gro-mov’s suggestion in the sense that the limits obtained in these papers have gener-alized nonnegative scalar curvature.Here we construct a sequence of Riemannian manifolds, M j , with positive scalarcurvature that converges in the intrinsic flat, metric measure and Gromov-Hausdor ff sense to a singular limit space, Y , which fails to satisfy (1) [Example 6.1]. In fact,the limit space is a sphere with a pulled thread:(2) Y = S / ∼ where a ∼ b i ff a , b ∈ C , where C is one geodesic in S (see Section 4). The scalar curvature about the point p = [ C ( t )] formed from the pulled thread is computed in Lemma 6.3 to be(3) lim r → Vol E ( B (0 , r )) − Vol M ( B ( p , r )) r Vol E ( B (0 , r )) = −∞ . In this sense the limit space does not have generalized nonnegative scalar curvature.We construct our sequence using a new method we call sewing developed inPropositions 3.1-3.3. Before we can sew the manifolds, the first two authors con-struct short tunnels between points in the manifolds building on prior work ofGromov-Lawson and Schoen-Yau in [GL80b] [SY79a]. The details of this con-struction are in the Appendix. In a subsequent paper [BS17] we will extend thissewing technique to also provide examples whose limit spaces fail to satisfy theScalar Torus Rigidity Theorem [SY79a] [GL80b] and the Positive Mass Rigid-ity Theorem [SY79b]. These examples, all constructed using the sewing tech-niques developed in this paper, demonstrate that Gromov-Hausdor ff and IntrinsicFlat limit spaces of noncollapsing sequences of manifolds with positive scalar cur-vature may fail to satisfy key properties of nonnegative scalar curvature.In light of these counter examples and the aforementioned positive results to-wards Gromov’s conjecture, the third author has suggested in [Sor17] to adapt the EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 3 class B . There it is proposed that the initial class of smooth Riemannian manifoldsin B should have nonnegative scalar curvature, a uniform lower bound on volume(as assumed implicitly by Gromov), and also a uniform lower bound on the min-imal area of a closed minimal surface in the manifold, MinA( M ). The sequencesof M j we construct using our new sewing methods have positive scalar curvatureand a uniform lower bound on volume, but MinA( M j ) →
0. Intuitive reasons as towhy a uniform lower bound on MinA( M j ) is a natural condition are described in[Sor17] along with a collection of related conjectures and open problems. Here wewill simply propose the following possible revision of Gromov’s vague conjecture: Conjecture 1.1.
Suppose a sequence of Riemannian manifolds, M j , have (4) Scal j ≥ , Vol( M j ) ≥ V > , and MinA ( M j ) ≥ A > , and the sequence converges in the intrinsic flat sense, M j F −→ M ∞ .Then at every point p ∈ M ∞ we have (5) lim r → Vol E ( B (0 , r )) − Vol Y ( B ( p , r )) r Vol E ( B (0 , r )) ≥ . This paper is part of the work towards Jorge Basilio’s doctoral dissertation at theCUNY Graduate Center conducted under the advisement of Professors J´ozef Dodz-iuk and Christina Sormani. We would like to thank Je ff Jauregui, Marcus Khuri,Sajjad Lakzian, Dan Lee, Raquel Perales, Conrad Plaut, and Catherine Searle fortheir interest in this work. 2. B ackground
In this section we first briefly review Gromov-Lawson and Schoen-Yau’s work.We then review Gromov-Hausdor ff , Metric Measure, and Intrinsic Flat Conver-gence covering the key definitions as well as theorems applied in this paper toprove our example converges with respect to all three notions of convergence.2.1. Gluing Gromov-Lawson and Schoen-Yau tunnels.
Using di ff erent tech-niques, Gromov-Lawson and Schoen-Yau described how to construct tunnels dif-feomorphic to S × [0 ,
1] with metric tensors of positive scalar curvature that canbe glued smoothly into three dimensional spheres of constant sectional curvature[GL80b][SY79a]. See Figure 1. These tunnels are the first crucial piece for ourconstruction. F igure
1. The Tunnel
J. BASILIO, J. DODZIUK, AND C. SORMANI
Here we need to explicitly estimate the volume and diameter of these tunnels.So the first and second authors prove the following lemma in the appendix.
Lemma 2.1.
Let < δ/ < . Given a complete Riemannian manifold, M ,that contains two balls B ( p i , δ/ ⊂ M , i = , , with constant positive sectionalcurvature K ∈ (0 , on the balls, and given any (cid:15) > , there exists a δ > su ffi ciently small so that we may create a new complete Riemannian manifold, N ,in which we remove two balls and glue in a cylindrical region, U, between them: (6) N = M \ ( B ( p , δ/ ∪ B ( p , δ/ (cid:116) Uwhere U = U ( δ ) has a metric of positive scalar curvature (See Figure 1) with (7) Diam( U ) ≤ h = h ( δ ) , where (8) h ( δ ) = O ( δ ) , hence, (9) lim δ → h ( δ ) = uniformly for K ∈ (0 , . The collars C i = B ( p i , δ/ \ B ( p i , δ ) identified with subsets of N have the originalmetric of constant curvature and the tunnel U (cid:48) = U \ ( C ∪ C ) has arbitrarilysmall diameter O ( δ ) and volume O ( δ ) . Therefore with appropriate choice of δ ,we have (10) (1 − (cid:15) )2 Vol( B ( p , δ/ ≤ Vol( U ) ≤ (1 + (cid:15) )2 Vol( B ( p , δ/ and (11) (1 − (cid:15) ) Vol( M ) ≤ Vol( N ) ≤ (1 + (cid:15) ) Vol( M ) . We note that if M has positive scalar curvature then so does N and that, afterinserting the tunnel, ∂ B ( p , δ/
2) and ∂ B ( p , δ/
2) are arbitrarily close together be-cause of (9). Note that we have restricted to three dimensions here and requiredconstant sectional curvature on the balls for simplicity. The first two authors willgeneralize these conditions in future work. This lemma su ffi ces for proving all theexamples in this paper.2.2. Review GH Convergence.
Gromov introduced the Gromov-Hausdor ff dis-tance in [Gro99].First recall that ϕ : X → Y is distance preserving i ff (12) d Y ( ϕ ( x ) , ϕ ( x )) = d X ( x , x ) ∀ x , x ∈ X . This is referred to as a metric isometric embedding in [LS14] and is distinct froma Riemannian isometric embedding.
Definition 2.2 (Gromov) . The Gromov-Hausdor ff distance between two compactmetric spaces ( X , d X ) and ( Y , d Y ) is defined as (13) d GH ( X , Y ) : = inf d ZH ( ϕ ( X ) , ψ ( Y )) EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 5 where Z is a complete metric space, and ϕ : X → Z and ψ : Y → Z are distancepreserving maps and where the Hausdor ff distance in Z is defined as (14) d ZH ( A , B ) = inf { (cid:15) > A ⊂ T (cid:15) ( B ) and B ⊂ T (cid:15) ( A ) } . Gromov proved that this is indeed a distance on compact metric spaces: d GH ( X , Y ) = ff there is an isometry between X and Y . When studying metric spaces whichare only precompact, one may take their metric completions before studying theGromov-Hausdor ff distance between them.We write(15) X j GH −→ X ∞ i ff d GH ( X j , X ∞ ) → . Gromov proved that if X j GH −→ X ∞ then there is a common compact metric space Z and distance preserving maps ϕ j : X j → Z such that(16) d ZH ( ϕ j ( X j ) , ϕ ∞ ( X ∞ )) → . We say p j ∈ X j converges to p ∞ ∈ X ∞ if there is such a set of maps such that ϕ j ( p j ) converges to ϕ ∞ ( p ∞ ) as points in Z . These limits are not uniquely definedbut they are useful and every point in the limit space is a limit of such a sequencein this sense. Theorem 2.3 (Gromov) . Suppose (cid:15) j → . If a sequence of metric spaces ( X j , d j ) have (cid:15) j almost isometries (17) F j : X j → X ∞ such that (18) | d ∞ ( F j ( p ) , F j ( q )) − d j ( p , q ) | ≤ (cid:15) j ∀ p , q ∈ X j and (19) X ∞ ⊂ T (cid:15) j ( F j ( X j )) then (20) X j GH −→ X ∞ . Note that p j ∈ X j converges to p ∞ ∈ X ∞ if F j ( p j ) → p ∞ ∈ X ∞ .Gromov’s Compactness Theorem states that a sequence of manifolds with non-negative Ricci (or Sectional) Curvature, and a uniform upper bound on diameter,has a subsequence which converges in the Gromov-Hausdor ff sense to a geodesicmetric space [Gro99]. If a sequence of manifolds has nonnegative sectional cur-vature, then they satisfy the Toponogov Triangle Comparison Theorem. Takingthe limits of the points in the triangles, one sees that the Gromov-Hausdor ff limitof the sequence also satisfies the triangle comparison. Thus the limit spaces areAlexandrov spaces with nonnegative Alexandrov curvature (cf. [BBI01]). J. BASILIO, J. DODZIUK, AND C. SORMANI
Review of Metric Measure Convergence.
Fukaya introduced the notion ofmetric measure convergence of metric measure spaces ( X j , d j , µ j ) in [Fuk87]. Heassumed the sequence converged in the Gromov-Hausdor ff sense as in (16) andthen required that the push forwards of the measures converge as well,(21) ϕ j ∗ µ j → ϕ ∞∗ µ ∞ weakly as measures in Z . Cheeger–Colding proved metric measure convergence of noncollapsing sequencesof manifolds with Ricci uniformly bounded below in [CC97] where the measureon the limit is the Hausdor ff measure. They proved metric measure convergence byconstructing almost isometries and showing the Hausdor ff measures of balls aboutconverging points converge:(22) If p j → p ∞ then H m ( B ( p j , r )) → H m ( B ( p ∞ , r )) . They also studied collapsing sequences obtaining metric measure convergence toother measures on the limit space. Cheeger and Colding applied this metric mea-sure convergence to prove that limits of manifolds with nonnegative Ricci curva-ture have generalized nonnegative Ricci curvature. In particular they prove thelimits satisfy the Bishop-Gromov Volume Comparison Theorem and the Cheeger-Gromoll Splitting Theorem.Sturm, Lott and Villani then developed the CD(k,n) notion of generalized Riccicurvature on metric measure spaces in [Stu06a][LV09]. In [Stu06b], Sturm ex-tended the study of metric measure convergence beyond the consideration of se-quences of manifolds which already converge in the Gromov-Hausdor ff sense,using the Wasserstein distance. This is also explored in Villani’s text [Vil09].CD(k,n) spaces converge in this sense to CD(k,n) spaces. RCD(k,n) spaces devel-oped by Ambrosio-Gigli-Savare are also preserved under this convergence [AGS14].RCD(k,n) spaces are CD(k,n) spaces which also require that the tangent cones al-most everywhere are Hilbertian. There has been significant work studying both ofthese classes of spaces proving they satisfy many of the properties of Riemannianmanifolds with lower bounds on their Ricci curvature.2.4. Review of Integral Current Spaces.
The Intrinsic Flat Distance is definedand studied in [SW11] by applying sophisticated ideas of Ambrosio-Kirchheim[AK00] extending earlier work of Federer-Fleming [FF60]. Limits of Riemann-ian manifolds under intrinsic flat convergence are integral current spaces, a notionintroduced by the third author and Stefan Wenger in [SW11].Recall that Federer-Flemming first defined the notion of an integral current asan extension of the notion of a submanifold of Euclidean space [FF60]. That isa submanifold ψ : M m → E N can be viewed as a current T = ψ [ M ] acting on m -forms as follows:(23) T ( ω ) = ψ [ M ]( ω ) = [ M ]( ψ ∗ ω ) = (cid:90) M ψ ∗ ω. EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 7 If ω = f d π ∧ · · · ∧ d π m then(24) T ( ω ) = ψ [ M ]( ω ) = (cid:90) M f ◦ ψ d ( π ◦ ψ ) ∧ · · · ∧ d ( π m ◦ ψ ) . They define boundaries of currents as ∂ T ( ω ) = T ( d ω ) so that then the boundaryof a submanifold with boundary is exactly what it should be. They define integerrectifiable currents more generally as countable sums of images under Lipschitzmaps of Borel sets. The integral currents are integer rectifiable currents whoseboundaries are integer rectifiable.Ambrosio-Kirchheim extended the notion of integral currents to arbitrary com-plete metric space [AK00]. As there are no forms on metric spaces, they use de-Georgi’s tuples of Lipschitz functions,(25) T ( f , π , ..., π m ) = ψ [ M ]( f , π , ..., π m ) = (cid:90) M f ◦ ψ d ( π ◦ ψ ) ∧ · · · ∧ d ( π m ◦ ψ ) . This integral is well defined because Lipschitz functions are di ff erentiable almosteverywhere. They define boundary as follows:(26) ∂ T ( f , π , ..., π m ) = T (1 , f , π , ..., π m )which matches with(27) d ( f d π ∧ · · · ∧ d π m ) = d f ∧ d π ∧ · · · ∧ d π m . They also define integer rectifiable currents more generally as countable sums ofimages under Lipschitz maps of Borel sets. The integral currents are integer recti-fiable currents whose boundaries are integer rectifiable.The notion of an integral current space was introduced in [SW11].
Definition 2.4.
An m dimensional integral current space, ( X , d , T ) , is a metricspace, ( X , d ) with an integral current structure T ∈ I m (cid:16) ¯ X (cid:17) where ¯ X is the metriccompletion of X and set(T) = X . Given an integral current space M = ( X , d , T ) wewill use set (M) or X M to denote X, d M = d and [ M ] = T . Note that set ( ∂ T) ⊂ ¯X .The boundary of ( X , d , T ) is then the integral current space: (28) ∂ ( X , d X , T ) : = (cid:0) set ( ∂ T) , d ¯X , ∂ T (cid:1) . If ∂ T = then we say ( X , d , T ) is an integral current without boundary. A compact oriented Riemannian manifold with boundary, M m , is an integralcurrent space, where X = M m , d is the standard metric on M and T is integrationover M . In this case M ( M ) = Vol( M ) and ∂ M is the boundary manifold. When M has no boundary, ∂ M = M ( T ) and the mass measure || T || of acurrent in [AK00]. We apply the same notions to define a mass for an integralcurrent space. Applying their theorems we have(29) M ( M ) = M ( T ) = (cid:90) X θ T ( x ) λ ( x ) d H m ( x )where λ ( x ) is the area factor and θ T is the weight. In particular λ ( x ) = x is Euclidean which is true on a Riemannian manifold where J. BASILIO, J. DODZIUK, AND C. SORMANI the weight is also 1. This is true almost everywhere in the examples in this paperas well. The mass measure, || T || , is a measure on X and satisfies(30) || T || ( A ) = (cid:90) A θ T ( x ) λ ( x ) d H m ( x ) . Review of the Intrinsic Flat distance.
The Intrinsic Flat distance was de-fined in work of the third author and Stefan Wenger [SW11] as a new distancebetween Riemannian manifolds based upon the Federer-Flemming flat distance[FF60] and the Gromov-Hausdor ff distance [Gro99].Recall that the Federer-Flemming flat distance between m dimensional integralcurrents S , T ∈ I m ( Z ) is given by(31) d ZF ( S , T ) : = inf { M ( U ) + M ( V ) : S − T = U + ∂ V } where U ∈ I m ( Z ) and V ∈ I m + ( Z ).In [SW11], the third author and Wenger imitate Gromov’s definition of theGromov-Hausdor ff distance (which he called the intrinsic Hausdor ff distance) byreplaced the Hausdor ff distance by the Flat distance: Definition 2.5. ( [SW11] ) For M = ( X , d , T ) and M = ( X , d , T ) ∈ M m letthe intrinsic flat distance be defined: (32) d F ( M , M ) : = inf d ZF ( ϕ T , ϕ T ) , where the infimum is taken over all complete metric spaces ( Z , d ) and distancepreserving maps ϕ : (cid:16) ¯ X , d (cid:17) → ( Z , d ) and ϕ : (cid:16) ¯ X , d (cid:17) → ( Z , d ) and the flatnorm d ZF is taken in Z. Here ¯ X i denotes the metric completion of X i and d i is theextension of d i on ¯ X i , while ϕ T denotes the push forward of T .
They then prove that this distance is 0 i ff the spaces are isometric with a currentpreserving isometry. They say(33) M j F −→ M ∞ i ff d F ( M j , M ∞ ) → . And prove that this happens i ff there is a complete metric space Z and distancepreserving maps ϕ j : M j → Z such that(34) d ZF ( ϕ j T j , ϕ ∞ T ∞ ) → Z hereis only complete and not compact.There is a special integral current space called the zero space,(35) = ( ∅ , , . Following the definition above, M j F −→ i ff d F ( M j , ) → Z and distance preserving maps ϕ j : M j → Z such that(36) d ZF ( ϕ j T j , → EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 9
Combining Gromov’s Embedding Theorem with Ambrosio-Kitrchheim’s Com-pactness Theorem one has:
Theorem 2.6 ([SW11]) . Given a sequence of m dimensional integral current spacesM j = (cid:16) X j , d j , T j (cid:17) such that X j are equibounded and equicompact and with uni-form upper bounds on mass and boundary mass. A subsequence converges inthe Gromov-Hausdor ff sense (cid:16) X j i , d j i (cid:17) GH −→ ( Y , d Y ) and in the intrinsic flat sense (cid:16) X j i , d j i , T j i (cid:17) F −→ ( X , d , T ) where either ( X , d , T ) is an m dimensional integral cur-rent space with X ⊂ Y or it is the current space. Note that in [SW10], the third author and Wenger prove if the M j have non-negative Ricci curvature then in fact the intrinsic flat and Gromov-Hausdor ff limitsagree. Matveev and Portegies have extended this to more general lower bounds onRicci curvature in [MP15]. With only lower bounds on scalar curvature the limitsneed not agree as seen in the Appendix of [SW11]. There are also sequences ofmanifolds with nonnegative scalar curvatue that have no Gromov-Hausdor ff limitbut do converge in the intrinsic flat sense (cf. Ilmanen’s Example presented in[SW11] and also [LS13]).In [Wen11], Wenger proved that any sequence of Riemannian manifolds with auniform upper bound on diameter, volume and boundary volume has a subsequencewhich converges in the intrinsic flat sense to an integral current space (cf. [SW11]).It is possible that the limit space is just the space which happens for examplewhen the volumes of the manifolds converge to 0.Note that when M j F −→ M ∞ the masses are lower semicontinuous:(37) lim inf j →∞ M ( M j ) ≥ M ( M ∞ )where the mass of an integral current space is just the mass of the integral currentstructure. The mass is just the volume when M is a Riemannian manifold andcan be computed using (29) otherwise. As there is not equality here, intrinsic flatconvergence does not imply metric measure convergence.In [Por15], Portegies has proven that when a sequence converges in the intrinsicflat sense and in addition M ( M j ) is assumed to converge to M ( M ∞ ), then the spacesdo converge in the metric measure sense, where the measures are taken to be themass measures.2.6. Useful Lemmas and Theorems concerning Intrinsic Flat convergence.
The following lemmas, definitions and theorems appear in work of the third author[Sor14], although a few (labelled only as c.f. [Sor14]) were used within proofs inolder work of the third author with Wenger [SW10]. All are proven rigorously in[Sor14].
Lemma 2.7. (c.f. [Sor14] ) A ball in an integral current space, M = ( X , d , T ) , withthe current restricted from the current structure of the Riemannian manifold is anintegral current space itself, (38) S ( p , r ) = (cid:0) set(T B(p , r)) , d , T B (cid:0) p , r (cid:1)(cid:1) for almost every r > . Furthermore, (39) B ( p , r ) ⊂ set(S(p , r)) ⊂ ¯B(p , r) ⊂ X . Lemma 2.8. (c.f. [Sor14] ) When M is a Riemannian manifold with boundary (40) S ( p , r ) = (cid:16) ¯ B ( p , r ) , d , T B ( p , r ) (cid:17) is an integral current space for all r > . Definition 2.9. (c.f. [Sor14] ) If M i = ( X i , d i , T i ) F −→ M ∞ = ( X ∞ , d ∞ , T ∞ ) , then wesay x i ∈ X i are a converging sequence that converge to x ∞ ∈ ¯ X ∞ if there exists acomplete metric space Z and distance preserving maps ϕ i : X i → Z such that (41) ϕ i T i F −→ ϕ ∞ T ∞ and ϕ i ( x i ) → ϕ ∞ ( x ∞ ) . If we say collection of points, { p , i , p , i , ... p k , i } , converges to a corresponding col-lection of points, { p , ∞ , p , ∞ , ... p k , ∞ } , if ϕ i ( p j , i ) → ϕ ∞ ( p j , ∞ ) for j = , . . . , k. Definition 2.10. (c.f. [Sor14] ) If M i = ( X i , d i , T i ) F −→ M ∞ = ( X ∞ , d ∞ , T ∞ ) , thenwe say x i ∈ X i are Cauchy if there exists a complete metric space Z and distancepreserving maps ϕ i : M i → Z such that (42) ϕ i T i F −→ ϕ ∞ T ∞ and ϕ i ( x i ) → z ∞ ∈ Z . We say the sequence is disappearing if z ∞ (cid:60) ϕ ∞ ( X ∞ ) . We say the sequence has nolimit in ¯ X ∞ if z ∞ (cid:60) ϕ ∞ ( ¯ X ∞ ) . Lemma 2.11. (c.f. [Sor14] ) If a sequence of integral current spaces, M i = ( X i , d i , T i ) ∈M m , converges to an integral current space, M = ( X , d , T ) ∈ M m , in the intrinsicflat sense, then every point x in the limit space X is the limit of points x i ∈ M i . Infact there exists a sequence of maps F i : X → X i such that x i = F i ( x ) converges tox and (43) lim i →∞ d i ( F i ( x ) , F i ( y )) = d ( x , y ) . Lemma 2.12. (c.f. [Sor14] ) If M j F −→ M ∞ and p j → p ∞ ∈ ¯ X ∞ , then for almostevery r ∞ > there exists a subsequence of M j also denoted M j such that (44) S ( p j , r ∞ ) = (cid:16) ¯ B (cid:16) p j , r ∞ (cid:17) , d j , T j B (cid:16) p j , r ∞ (cid:17)(cid:17) are integral current spaces for j ∈ { , , ..., ∞} and we have (45) S ( p j , r ∞ ) F −→ S ( p ∞ , r ∞ ) . If p j are Cauchy with no limit in ¯ X ∞ then there exists δ > such that for almostevery r ∈ (0 , δ ) such that S ( p j , r ) are integral current spaces for j ∈ { , , ... } andwe have (46) S ( p j , r ) F −→ . If M j F −→ then for almost every r and for all sequences p j we have (46). EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 11
Theorem 2.13. (c.f. [Sor14] ) Suppose M i = ( X i , d i , T i ) are integral current spacesand (47) M i F −→ M ∞ , and suppose we have Lipschitz maps into a compact metric space Z, (48) F i : X i → Z with
Lip( F i ) ≤ K , then a subsequence converges to a Lipschitz map (49) F ∞ : X ∞ → Z with
Lip( F ∞ ) ≤ K . More specifically, there exists distance preserving maps of the subsequence, ϕ i : X i → Z, such that (50) d ZF ( ϕ i T i , ϕ ∞ T ∞ ) → and for any sequence p i ∈ X i converging to p ∈ X ∞ (i.e. d Z ( ϕ i ( p i ) , ϕ ∞ ( p )) → ),we have (51) lim i →∞ F i ( p i ) = F ∞ ( p ∞ ) . Theorem 2.14. (c.f. [Sor14] ) Suppose M mi = ( X i , d i , T i ) are integral current spaceswhich converge in the intrinsic flat sense to a nonzero integral current space M m ∞ = ( X ∞ , d ∞ , T ∞ ) . Suppose there exists r > and a sequence p i ∈ M i such that foralmost every r ∈ (0 , r ) we have integral current spaces, S ( p i , r ) , for all i ∈ N and (52) lim inf i →∞ d F ( S ( p i , r ) , ) = h > . Then there exists a subsequence, also denoted M i , such that p i converges to p ∞ ∈ ¯ X ∞ . Theorem 2.15. (c.f. [Sor14] ) Let M i = ( X i , d i , T i ) and M (cid:48) i = ( X (cid:48) i , d (cid:48) i , T i ) be integralcurrent spaces with (53) M ( M i ) ≤ V and M ( ∂ M i ) ≤ A such that (54) M i F −→ M ∞ and M (cid:48) i F −→ M (cid:48)∞ . Fix δ > . Let F i : M i → M (cid:48) i be continuous maps which are isometries on ballsof radius δ : (55) ∀ x ∈ X i , F i : ¯ B ( x , δ ) → ¯ B ( F i ( x ) , r ) is an isometryThen, when M ∞ (cid:44) , we have M (cid:48)∞ (cid:44) and there is a subsequence, also denotedF i , which converges to a (surjective) local current preserving isometry (56) F ∞ : ¯ X ∞ → ¯ X (cid:48)∞ satisfying (55) . More specifically, there exists distance preserving maps of the subsequence ϕ i : X i → Z, ϕ (cid:48) i : X (cid:48) i → Z (cid:48) , such that (57) d ZF ( ϕ i T i , ϕ ∞ T ∞ ) → and d Z (cid:48) F ( ϕ (cid:48) i T (cid:48) i , ϕ (cid:48)∞ T (cid:48)∞ ) → and for any sequence p i ∈ X i converging to p ∈ X ∞ : (58) lim i →∞ ϕ i ( p i ) = ϕ ∞ ( p ) ∈ Zwe have (59) lim i →∞ ϕ (cid:48) i ( F i ( p i )) = ϕ (cid:48)∞ ( F ∞ ( p ∞ )) ∈ Z (cid:48) . When M ∞ = and F i are surjective, we have M (cid:48)∞ = .
3. S ewing R iemannian M anifolds with P ositive S calar C urvature The main technique we will introduce in this paper is the construction of threedimensional manifolds with positive scalar curvature through a process we call“sewing” which involved gluing a sequence of tunnels along a curve. We applyLemma 2.1 which constructs Gromov-Lawson Schoen-Yau tunnels. The lemma isproven in the Appendix.3.1.
Gluing Tunnels between Spheres.
We begin by gluing tunnels between ar-bitrary collections of pairs of spheres as in Figure 2.
Proposition 3.1.
Given a complete Riemannian manifold, M , and A ⊂ M acompact subset with an even number of points p i ∈ A , i = , . . . , n, with pairwisedisjoint contractible balls B ( p i , δ ) which have constant positive sectional curvatureK, for some δ > , define A δ = T δ ( A ) and (60) A (cid:48) δ = A δ \ n (cid:91) i = B ( p i , δ/ (cid:116) n / (cid:91) i = U i where U i are the tunnels as in Lemma 2.1 connecting ∂ B ( p j + , δ/ to ∂ B ( p j + , δ/ for j = , , . . . , n / − . Then given any (cid:15) > , shrinking δ further, if necessary,we may create a new complete Riemannian manifold, N , (61) N = ( M \ A δ ) (cid:116) A (cid:48) δ satisfying (62) (1 − (cid:15) ) Vol( A δ ) ≤ Vol( A (cid:48) δ ) ≤ Vol( A δ )(1 + (cid:15) ) and (63) (1 − (cid:15) ) Vol( M ) ≤ Vol( N ) ≤ Vol( M )(1 + (cid:15) ) . If, in addition, M has non-negative or positive scalar curvature, then so doesN . In fact, (64) inf x ∈ M Scal x ≥ min (cid:40) , inf x ∈ N Scal x (cid:41) If ∂ M (cid:44) ∅ , the balls avoid the boundary and ∂ M is isometric to ∂ N . Definition 3.2.
We say that we have glued the manifold to itself with a tunnelbetween the collection of pairs of sphere ∂ B ( p i , δ ) to ∂ B ( p i + , δ ) for i = to n − .See Figure 2. EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 13 F igure
2. Gluing two spheres with a tunnel.
Proof.
For simplicity of notation, set A = A δ and A (cid:48) = A (cid:48) δ .By induction on n and Lemma 2.1, we see that N can be given a metric ofpositive scalar curvature whenever M has positive scalar curvature.Using the fact that the balls are pairwise disjoint and of the same volume, and(10) from Lemma 2.1, we have the volume of A (cid:48) can be estimated:Vol( A (cid:48) ) = Vol( A ) − n (cid:88) i = Vol( B ( p i , δ/ + n / (cid:88) i = Vol( U i ) = Vol( A ) + n · (Vol( U i ) − B ( p i , δ/ ≤ Vol( A ) + n · (2 Vol( B ( p i , δ/ · (cid:15) ) = Vol( A ) + (cid:15) · ( n Vol( B ( p i , δ/ ≤ Vol( A ) + (cid:15) Vol( A )which yields the right-hand side of (62).Similarly, Vol( A (cid:48) ) = Vol( A ) − n (cid:88) i = Vol( B ( p i , δ/ + n / (cid:88) i = Vol( U i ) = Vol( A ) + n · (Vol( U i ) − B ( p i , δ/ ≥ Vol( A ) + n · ( − B ( p i , δ/ · (cid:15) ) = Vol( A ) − (cid:15) · ( n Vol( B ( p i , δ/ ≥ Vol( A ) − (cid:15) Vol( A )which yields the left-hand side of (62). To estimate the volume of N we will use the volume estimates for A (cid:48) . Using(10) from Lemma 2.1 again, we haveVol( N ) = Vol( M ) − Vol( A ) + Vol( A (cid:48) ) ≤ Vol( M ) − Vol( A ) + (1 + (cid:15) ) Vol( A ) = Vol( M ) + (cid:15) Vol( A ) (by (11)) ≤ Vol( M ) + (cid:15) Vol( M ) , which yields the right-hand side of (63).Similarly, Vol( N ) = Vol( M ) − Vol( A ) + Vol( A (cid:48) ) ≥ Vol( M ) − Vol( A ) + (1 − (cid:15) ) Vol( A ) = Vol( M ) − (cid:15) Vol( A ) (by (11)) ≥ Vol( M ) − (cid:15) Vol( A ) , which yields the left-hand side of (63).Finally, observe that (64) follows since Lemma 2.1 shows that the tunnels U i have positive scalar curvature. (cid:3) Sewing along a Curve.
We now describe our process we call sewing along acurve, where a sequence of balls is taken to be located along curve much like holescreated when stitching a thread. We glue a sequence of tunnels to the boundariesof these balls as in Figure 3. We say that we have sewn the manifold along thecurve C through the given balls. By gluing tunnels in this precise way we are ableto shrink the diameter of the edited tubular neighborhood around the curve becausetravel along the curve can be conducted e ffi ciently through the tunnels.F igure
3. Sewing a manifold through eight balls along a curve.
Proposition 3.3.
Given a complete Riemannian manifold, M , and A ⊂ M Riemannian isometric to an embedded curve, C : [0 , → S K possibly withC (0) = C (1) and parametrized proportional to arclength, in a standard sphere EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 15 of constant sectional curvature K, define A a = T a ( A ) as in Proposition 3.1 andassume that A a is Riemannian isometric to T a ( C ) ⊂ S K . Then, given any (cid:15) > there exists n su ffi ciently large and δ = δ ( (cid:15), n , C , K ) > su ffi ciently small as in(66) so that we can “sew along the curve” to create a new complete Riemannianmanifold N , (65) N = ( M \ A δ ) (cid:116) A (cid:48) δ , exactly as in Proposition 3.1, for (66) δ = δ ( (cid:15), n , C , K ) such that δ < a , lim n →∞ n · h ( δ ) = , and lim n →∞ n · δ = , where h is defined in Lemma 2.1 and the disjoint balls B ( p i , δ ) are to be centeredat (67) p j + = C (cid:32) jn + δ L ( C ) (cid:33) p j + = C (cid:32) j + n − δ L ( C ) (cid:33) j = , , . . . , n − and (68) A (cid:48) δ = A δ \ n (cid:91) i = B ( p i , δ/ (cid:116) n − (cid:91) j = U j + . Thus, the tunnels U j + connect ∂ B ( p j + , δ ) to ∂ B ( p j + , δ ) for j = , , . . . , n − .Furthermore, (69) (1 − (cid:15) ) Vol( A δ ) ≤ Vol( A (cid:48) δ ) ≤ Vol( A δ )(1 + (cid:15) ) and (70) (1 − (cid:15) ) Vol( M ) ≤ Vol( N ) ≤ Vol( M )(1 + (cid:15) ) and (71) Diam( A (cid:48) δ ) ≤ H ( δ ) = L ( C ) / n + ( n + h ( δ ) + (5 n + δ. Since (72) lim δ → H ( δ ) = uniformly for K ∈ (0 , , we say we have sewn the curve, A , arbitrarily short.If, in addition, M has non-negative or positive scalar curvature, then so doesN . In fact, (73) inf x ∈ M Scal x ≥ min (cid:40) , inf x ∈ N Scal x (cid:41) If ∂ M (cid:44) ∅ , the balls avoid the boundary and ∂ M is isometric to ∂ N .Proof. By the fact that C is embedded, for n su ffi ciently large, the balls in thestatement are disjoint even when C (0) = C (1) so we may apply Propositon 3.1 toget (69) and (70).For simplicity of notation, let A = A δ and A (cid:48) = A (cid:48) δ . We now verify the diameter estimate of A (cid:48) , (71). To do this we define sets C i ⊂ A (cid:48) which correspond to the sets ∂ B ( p i , δ/ ⊂ A which are unchanged becausethey are the boundaries of the edited regions:(74) C i ∪ C i + = ∂ U i , whenever i is an odd value. Let(75) U = n − (cid:91) j = U j + . Let x and y be arbitrary points in A (cid:48) . We claim that there exists j , k ∈ { , . . . , n } such that(76) d A (cid:48) ( x , C j ) < δ + L ( C ) / (2 n ) + h ( δ ) and d A (cid:48) ( y , C k ) < δ + L ( C ) / (2 n ) + h ( δ )By symmetry we need only prove this for x . Note that in case I where(77) x ∈ A (cid:48) \ U = A \ n (cid:91) i = B ( p i , δ/ x as a point in A . Let γ ⊂ A be the shortest path from x to the closestpoint c x ∈ C [0 ,
1] so that L ( γ ) < δ .If(78) γ ∩ B ( p j , δ/ (cid:44) ∅ then(79) d A (cid:48) \ U ( x , C j ) < δ and we have that (76) holds. Otherwise, still in Case I, if (78) fails then we have d A (cid:48) \ U ( x , C j ) ≤ d A (cid:48) \ U ( x , c x ) + d ( c x , C j ) (by the triangle inequality)(80) < δ + L ( C )2 n , (81)where the last inequality follows from d A (cid:48) \ U ( x , c x ) ≤ L ( γ ) < δ and the fact that c x ∈ C ([0 , L ( C ) / (2 n ) away from the boundary of the nearest tunnel.Alternatively, we have case II where x ∈ U . In this case, there exists j such that x ∈ U j + and so(82) d A (cid:48) ( x , C j + ) ≤ Diam( U j + ) ≤ h ( δ ) . Thus, we have the claim in (76).We now proceed to prove (71) by estimating d A (cid:48) ( x , y ) for x , y ∈ A (cid:48) . If j = k in(76), then d A (cid:48) ( x , y ) ≤ δ + L ( C ) / (2 n ) + h ( δ )) and we are done. Otherwise, by (76)and the triangle inequality, we have d A (cid:48) ( x , y ) ≤ d A (cid:48) ( x , C j ) + d A (cid:48) ( y , C k ) + sup { d A (cid:48) ( z , w ) | z ∈ C j , w ∈ C k } (83) ≤ δ + L ( C ) / (2 n ) + h ( δ )) + sup { d A (cid:48) ( z , w ) | z ∈ C j , w ∈ C k } . (84) EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 17
Without loss of generality, we may assume that j < k and that j is odd. Thus, C j ⊂ ∂ U j . If k is also odd then by the triangle inequalitysup { d A (cid:48) ( z , w ) | z ∈ C j , w ∈ C k } ≤ Diam( U j ) + dist( U j , U j + )(85) + Diam( U j + ) + · · · + Diam( U k − ) + dist( U k − , U k )and, when k is even,sup { d A (cid:48) ( z , w ) | z ∈ C j , w ∈ C k } ≤ Diam( U j ) + dist( U j , U j + )(86) + Diam( U j + ) + · · · + Diam( U k − ) + dist( U k − , U k − ) + Diam( U k − ) . We know that Diam( U j ) = · · · = Diam( U k ) ≤ h ( δ ) from (7) of Lemma 2.1, andthat the distance between any two adjacent tunnels is the same, and that there areat most n tunnels. Thus, in either case (85) or (86) we have(87) sup { d A (cid:48) ( z , w ) | z ∈ C j , w ∈ C k } ≤ n h ( δ ) + n · dist( U j , U j + ) . and by construction the distance between adjacent tunnels isdist( U j , U j + ) ≤ Diam( C j + ) + dist( C j + , C j + ) + Diam( C j + )(88) ≤ π ( δ/ + δ + π ( δ/ < δ (89)since the balls B ( p i , δ/
2) have constant sectional curvature K .Therefore, combining (84), (87) and (89) we conclude that(90) d A (cid:48) ( x , y ) ≤ δ + L ( C ) / (2 n ) + h ( δ )) + n h ( δ ) + n δ which is the desired diameter estimate (71).We observe that by our choice of δ satisfying (66) and the fact that h ( δ ) = O ( δ )from Lemma 2.1 we have that (72) holds.Finally, observe that (73) follows since Lemma 2.1 shows that the tunnels U i have positive scalar curvature. (cid:3)
4. P ulled S tring S paces The following notion of a pulled string metric space captures the idea that ifa metric space is a patch of cloth and a curve in the patch is sewn with a string,then one can pull the string tight, identifying the entire curve as a single point, thuscreating a new metric space. This notion was first described to the third author byBurago when they were working ideas related to [BI09]. See Figure 4.
Proposition 4.1.
The notion of a metric space with a pulled string is a metric space ( Y , d Y ) constructed from a metric space ( X , d X ) with a curve C : [0 , → X, so that (91) Y = X \ C [0 , (cid:116) { p } , p = C (0) , where for x i ∈ Y we have (92) d Y ( x , p ) = min { d X ( x , C ( t )) : t ∈ [0 , } and for x i ∈ X \ C [0 , we have (93) d Y ( x , x ) = min { d X ( x , x ) , min { d X ( x , C ( t )) + d X ( x , C ( t )) : t i ∈ [0 , } } . If ( X , d , T ) is a Riemannian manifold then ( Y , d , ψ T ) is an integral current spacewhose mass measure is the Hausdor ff measure on Y and (94) H mY ( Y ) = H mX ( X ) − H mX ( K ) . If ( X , d X , T ) is an integral current space then ( Y , d Y , ψ T ) is also an integral currentspace where ψ : X → Y such that ψ ( x ) = x for all x ∈ X \ C [0 , and ψ ( C ( t )) = p for all t ∈ [0 , . So that (95) M ( ψ T ) = M ( T )F igure
4. A two sphere with the equator pulled to a point.We will in fact prove this proposition as a consequence of two lemmas aboutspaces with arbitrary compact subsets pulled to a point. Lemma 4.2 proves such aspace is a metric space and Lemma 4.3 proves (94) and (95).4.1.
Pulled string spaces are metric spaces.Lemma 4.2.
Given a metric space ( X , d X ) and a compact set K ⊂ X we may definea new metric space ( Y , d Y ) by pulling the set K to a point p ∈ K by setting (96) Y : = X \ K (cid:116) { p } , p ∈ K fixed , and, for x ∈ Y, we have (97) d Y ( x , p ) = min { d X ( x , y ) : y ∈ K } and, for x i ∈ Y \ { p } , we have (98) d Y ( x , x ) = min { d X ( x , x ) , min { d X ( x , y ) + d X ( x , y ) : y i ∈ K }} . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 19
Proof.
We first prove that ( Y , d Y ) is a metric space. By definition, it is easy to seethat d Y is non-negative and symmetric. To prove that d Y satisfies the axiom of posi-tivity, assume x = x . Then either x i = p , and d Y ( x , x ) = x i (cid:44) p and d X ( x , x ) = d Y ( x , x ) ≤ d X ( x , x ) = d Y ( x , x ) = d X ( x , x ) = = min { d X ( x , y ) + d X ( x , y ) | y i ∈ K } . In the first case, x = x since d X is a metric, so assume otherwise. Then d X ( x , x ) (cid:44) d X ( x , y ) = d X ( x , y ) = y i ∈ K . Hence, x i = y i which is impos-sible by the definition of Y unless x = x = p which yields a contradiction. Thisproves that d Y satisfies positivity.Next, let us note that by virtue of (97) and (98), we always have(100) d Y ( x , x ) ≤ d X ( x , x ) , ∀ x , x ∈ Y and(101) if d Y ( x , x ) (cid:44) d X ( x , x ) = ⇒ d Y ( x , x ) = d X ( x , y ) + d X ( x , y ) . for some y i ∈ K .We now verify the triangle inequality: for any x , x , x ∈ Y , we need to prove(102) d Y ( x , x ) ≤ d Y ( x , x ) + d Y ( x , x ) . It will be convenient to define y i ∈ K such that(103) d X ( x i , y i ) = min { d X ( x i , y ) | y ∈ K } for i = , , . Assume in Case I that d Y ( x , x ) (cid:44) d X ( x , x ). Then by (101) and (103),(104) d Y ( x , x ) = d X ( x , y ) + d X ( x , y ) . We have three possibilities: (i) d Y ( x , x ) (cid:44) d X ( x , x ) and d Y ( x , x ) (cid:44) d X ( x , x );(ii) d Y ( x , x ) = d X ( x , x ) and d Y ( x , x ) = d X ( x , x ); and (iii) (without loss ofgenerality) d Y ( x , x ) (cid:44) d X ( x , x ) and d Y ( x , x ) = d Y ( x , x ).In Case I (i), we have d Y ( x , x ) = d X ( x , y ) + d X ( x , y ) (by (104)) ≤ d X ( x , y ) + d X ( x , y ) + d X ( x , y ) + d X ( x , y ) = d Y ( x , x ) + d Y ( x , x ) . (by assumption (i), (101), and (103))In Case I (ii), we have d Y ( x , x ) ≤ d X ( x , x ) (by (100)) ≤ d X ( x , x ) + d X ( x , x ) = d Y ( x , x ) + d Y ( x , x ) . (by assumption (ii)) In Case I (iii), we have d X ( x , y ) = min { d X ( x , K ) | y ∈ K } (by (103)) ≤ d X ( x , y ) ≤ d X ( x , x ) + d X ( x , y )(105) ≤ d Y ( x , x ) + d X ( x , y ) (by assumption (iii))(106)so that d Y ( x , x ) = d X ( x , y ) + d X ( x , y ) (by (104)) ≤ d X ( x , y ) + d Y ( x , x ) + d X ( x , y ) (by (106)) = d Y ( x , x ) + d Y ( x , x ) . (by assumption (iii))This proves the triangle inequality, (102), in Case I. Next, we assume, in Case II,that d Y ( x , x ) = d X ( x , x ).Again, we have three possibilities: (i) d Y ( x , x ) (cid:44) d X ( x , x ) and d Y ( x , x ) (cid:44) d X ( x , x ); (ii) d Y ( x , x ) = d X ( x , x ) and d Y ( x , x ) = d X ( x , x ); and (iii) (with-out loss of generality) d Y ( x , x ) (cid:44) d X ( x , x ) and d Y ( x , x ) = d Y ( x , x ).In Case II (i), we have d Y ( x , x ) = d X ( x , x ) ≤ d X ( x , y ) + d X ( x , y ) (by (104)) ≤ d X ( x , y ) + d X ( x , y ) + d X ( x , y ) + d X ( x , y ) = d Y ( x , x ) + d Y ( x , x ) . (by assumption (i), (101), and (103))In Case II (ii), (102) follows immediately from the triangle inequality for d X .Finally, in Case II (iii), d Y ( x , x ) = d X ( x , x ) ≤ d X ( x , y ) + d X ( x , y ) (by (104)) ≤ d X ( x , y ) + d X ( x , x ) + d X ( x , y ) = d Y ( x , x ) + d Y ( x , x ) , (by assumption (iii), (101), and (103))which completes the proof. (cid:3) Hausdor ff Measures and Masses of Pulled String Spaces.Lemma 4.3. If ( X , d X , T ) is an integral current space with a compact subset K ⊂ Xthen ( Y , d Y , ψ T ) is also an integral current space where ( Y , d Y ) is defined as inLemma 4.2 and where ψ : X → Y such that ψ ( x ) = x for all x ∈ X \ K and ψ ( q ) = p for all q ∈ K. In addition (107) M ( ψ T ) = M ( T ) − || T || ( K ) If ( X , d X , T ) is a Riemannian manifold then ( Y , d Y , ψ T ) is an integral current spacewhose mass measure is the Hausdor ff measure on Y and (108) H mY ( Y ) = H mX ( X ) − H mX ( K ) . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 21
Proof.
Next, suppose that ( X , d X , T ) is an m -dimensional integral current space.We must show that ( Y , d Y , ψ T ) is an integral current space. We first observethat ψ as defined in the statement of the proposition is a 1-Lipschitz function:for x , y ∈ X \ K , there is no ambiguity so we may view them as elements of Y \ { p } and d Y ( ψ ( x ) , ψ ( y )) = d Y ( x , y ) ≤ d X ( x , y ) by definition of d Y . Otherwise,we may assume, without loss of generality, that x ∈ K and y (cid:60) K . In this case, d Y ( ψ ( x ) , ψ ( y )) = d Y ( p , ψ ( y )) = d Y ( p , y ) = min { d X ( z , y ) : z ∈ K } ≤ d X ( x , y ), as x ∈ K . Thus, ψ T is an integral current on Y since ψ is a 1-Lipschitz function andthe well-known inequality(109) (cid:107) ψ T (cid:107) ≤ Lip( ψ ) m (cid:107) T (cid:107) implies that ψ T has finite mass because T does. To show that ( Y , d Y , ψ T ) is anintegral current space there remains to show that it is completely settled, or ψ T has positive density at p .Let f : Y → R be a bounded Lipschitz map and π j : Y → R be Lipschitz maps.Then( ψ T )( f , π , . . . , π m ) = T ( f ◦ ψ, π ◦ ψ, . . . , π m ◦ ψ ) = T ( f · X \ K + f ( p ) · K , π ◦ ψ, . . . , π m ◦ ψ ) = T ( f · X \ K , π ◦ ψ, . . . , π m ◦ ψ ) + f ( p ) T (1 K , π ◦ ψ, . . . , π m ◦ ψ ) = T ( f · X \ K , π ◦ ψ, . . . , π m ◦ ψ ) + π i ◦ ψ are constant on { K (cid:44) } (see [AK00]) so( ψ T )( f , π , . . . , π m ) = T ( f · X \ K , π ◦ ψ, . . . , π m ◦ ψ ) = ( T X \ K )( f , π ◦ ψ, . . . , π m ◦ ψ ) = ( T X \ K )( f ◦ ψ, π ◦ ψ, . . . , π m ◦ ψ )because ψ ( x ) = x on X \ K , = ψ ( T X \ K )( f , π , . . . , π m ) . So, using the characterization of mass from [AK00], (2.6) of Proposition 2.7, M ( ψ T ) = M ( ψ ( T X \ K )) = M ( T X \ K )because ψ ( x ) = x on X \ K , so since M ( · ) = (cid:107) · (cid:107) ,( ψ T )( f , π , . . . , π m ) = (cid:107) T X \ K (cid:107) ( X ) = sup ∞ (cid:88) j = | ( T X \ K )(1 A j , π j , . . . , π jm ) | , where the supremum is taken over all Borel partitions { A j } of X such that X = ∪ j A j and all Lipschitz functions π ji ∈ Lip( X ) with Lip( π ji ) ≤
1, then continuing( ψ T )( f , π , . . . , π m ) = sup ∞ (cid:88) j = | T (1 X \ K · A j , π j , . . . , π jm ) | = sup ∞ (cid:88) j = | T (1 ˜ A j , ˜ π j , . . . , ˜ π jm ) | , where the second supremum is taken over all Borel partitions { ˜ A j } of X \ K suchthat X \ K = ∪ j ˜ A j and all Lipschitz functions ˜ π ji ∈ Lip( X \ K ) with Lip( ˜ π ji ) ≤ ψ T )( f , π , . . . , π m ) = sup ∞ (cid:88) j = | T (1 ˜ A j , ˜ π j , . . . , ˜ π jm ) | = || T || ( X \ K ) = || T || ( X ) − || T | ( K ) = M ( T ) − || T || ( K ) , which proves (107).Finally, assume that the m -dimensional integral current space ( X , d X , T ) is a Rie-mannian manifold. We show that the mass measure of ( Y , d Y , ψ T ) is the Hausdor ff measure on ( Y , d Y ).We claim that(110) H mY ( Y \ { p } ) = H mX ( X \ K ) . First, observe that since ψ is 1-Lipschitz, H mY ( ψ ( X \ K )) ≤ (Lip( ψ )) m H mX ( X \ K ) , by Proposition 3.1.4 on page 37 from [AT04], hence H mY ( Y \ { p } ) ≤ H mX ( X \ K ) . Thus, there remains to show the opposite inequality in (110).Define sets C j = { y ∈ Y | d Y ( y , p ) ≥ / j } for each j ∈ N . Then the C j are closed sets, C j ⊂ C j + and Y \ { p } = ∪ j ∈ N C j . Sowe may use Theorem 1.1.18 from [AT04]:(111) H mY ( Y \ { p } ) = H mY ( ∪ j ∈ N C j ) = lim j →∞ H mY ( C j ) . Consider, for each j ∈ N , D j = ψ − ( C j ) = { x ∈ X | d X ( x , K ) ≥ / j } EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 23 which are closed in X , D j ⊂ D j + , and X \ K = ∪ j ∈ N D j . Using Theorem 1.1.8 from[AT04] again:(112) H mX ( X \ K ) = H mX ( ∪ j ∈ N D j ) = lim j →∞ H mX ( D j ) . Next, we claim that(113) H mX ( D j ) ≤ H mY ( C j ) , j ∈ N . Fix j . Fix δ < j . Let { E l } l ∈ N be a countable cover of C j with Diam( E l ) < δ , for all l . Then(114) dist( E l , p ) > j , l ∈ N . To see this, assume otherwise. Then since dist Y ( p , E l ) < j and the definition ofdistance (as an infimum), there is e ∈ E l such that d Y ( p , e ) < j . Now, we alsoknow that E l ∩ C j (cid:44) ∅ . So, there is c ∈ C j ∩ E l . So, d Y ( e , c ) ≤ Diam Y ( E l ) < δ < j .Also, by the triangle inequality, d Y ( p , c ) ≤ d Y ( p , e ) + d Y ( e , c ) < / j . But thiscontradicts that c ∈ C j as by definition of C j , d Y ( p , c ) > / j .Next, we show that(115) Diam Y ( E l ) = Diam X ( ψ − ( E l )) , i.e. ψ − is an isometry when restricted to { E l } . In fact, we prove d X ( ψ − ( a ) , ψ − ( b )) = d Y ( a , b ) , ∀ a , b ∈ E l , j ∈ N . Let a , b ∈ E l . Then since Diam( E l ) < δ < j we have d Y ( a , b ) ≤ Diam Y ( E l ) <δ < j , so(116) d Y ( a , b ) < j . By definition of the distance d Y , since ψ − ( a ) = a and ψ − ( b ) = b , d Y ( a , b ) = min (cid:8) d X ( a , b ) , min { d X ( a , k ) + d X ( b , k ) | k i ∈ K } (cid:9) . If d Y ( a , b ) = d X ( a , b ), we’re done. If not, then there exists k , k ∈ K so that(117) d Y ( a , b ) = d X ( a , k ) + d X ( b , k ) . By (114), d Y ( a , p ) ≥ j and d Y ( b , p ) ≥ j which implies dist X ( a , K ) ≥ j and dist X ( b , K ) ≥ j . But then 1 j ≤ dist X ( a , K ) + dist X ( b , K ) ≤ d X ( a , k ) + d X ( b , k ) = d X ( a , b ) (by (117)) < j , (by (116))which is a contradiction.Next, observe that { ψ − ( E l ) } l ∈ N is necessarily a cover of D j so H mX ( D j ) ≤ ∞ (cid:88) l = ω m (cid:32) Diam X ( ψ − ( E l ))2 (cid:33) m = ∞ (cid:88) l = ω m (cid:32) Diam Y ( E l )2 (cid:33) m . (by (115))Taking the infimum over all covers of C j with diameters less than δ gives H mX ( D j ) ≤ H mY ,δ ( C j )then taking the limit as δ → H mX ( D j ) ≤ H mY ( C j )which proves the claim (113).To finish, we take the limit in (113) as j → ∞ and use (111) and (112) tocomplete the proof. (cid:3)
5. S ewn M anifolds converging to P ulled S trings In this section we consider a sequences of sewn manifolds being sewn increas-ingly tightly and prove they converge in the Gromov-Hausdor ff and Intrinsic Flatsense to metric spaces with pulled strings.To be more precise, we consider the following sequences of increasingly tightlysewn manifolds: Definition 5.1.
Given a single Riemannian manifold, M , with a curve, A = C ([0 , ⊂ M, with a tubular neighborhood A = T a ( A ) which is Riemannian iso-metric to a tubular neighborhood of a compact set V ⊂ S K , in a standard sphereof constant sectional curvature K, satisfying the hypothesis of Proposition 3.3. Wecan construct its sequence of increasingly tightly sewn manifolds, N j , by applyingProposition 3.3 taking (cid:15) = (cid:15) j → , n = n j → ∞ , and δ = δ j → to create eachsewn manifold, N = N j and the edited regions A (cid:48) δ = A (cid:48) δ j which we simply denoteby A (cid:48) j . This is depicted in Figure 5. Since these sequences N j are created us-ing Proposition 3.3, they have positive scalar curvature whenever M has positivescalar curvature, and ∂ N j = ∂ M whenever M has a nonempty boundary. EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 25 F igure
5. A sequence of increasingly tightly sewn manifolds.In this section we prove Lemma 5.5, Lemma 5.6 and Lemma 5.7, which imme-diately imply the following theorem:
Theorem 5.2.
The sequence N j as in Definition 5.1 converges in the Gromov-Hausdor ff sense (118) N j GH −→ N ∞ , the metric measure sense (119) N j mGH −→ N ∞ , and the intrinsic flat sense (120) N j F −→ N ∞ , where N ∞ is the metric space created by pulling the string, A = C ([0 , ⊂ M, toa point as in Proposition 4.1.
In fact our lemmas concern more general sequences of manifolds which areconstructed from a given manifold M and scrunch a given compact set K ⊂ M down to a point as follows: Definition 5.3.
Given a single Riemannian manifold, M , with a compact set, A ⊂ M. A sequence of manifolds, (121) N j = ( M \ A δ j ) (cid:116) A (cid:48) δ j is said to scrunch A down to a point if A δ = T δ ( A ) and A (cid:48) δ satisfies: (122) (1 − (cid:15) ) Vol( A δ ) ≤ Vol( A (cid:48) δ ) ≤ Vol( A δ )(1 + (cid:15) ) and (123) (1 − (cid:15) ) Vol( M ) ≤ Vol( N ) ≤ Vol( M )(1 + (cid:15) ) and (124) Diam( A (cid:48) δ ) ≤ Hwhere (cid:15) = (cid:15) j → and where H = H j → and δ j < H j . Note that by Proposition 3.3, a sequence of increasingly tightly sewn manifoldssewn along a curve C ([0 , A = C ([0 , Constructing Surjective maps to the limit spaces.
Before we prove con-vergence of the scrunched sequence of manifolds to the pulled thread space, weconstruct surjective maps from the sequence to the proposed limit space.
Lemma 5.4.
Given M a compact Riemannian manifold (possibly with boundary)and a smooth embedded compact zero to three dimensional submanifold A ⊂ M (possibly with boundary), and N j as in Definition 5.3. Then for j su ffi ciently largethere exist surjective Lipschitz maps (125) F j : N j → N ∞ with Lip( F j ) ≤ where N ∞ is the metric space created by taking M and pulling A to a point p asin Lemmas 4.2- 4.3. Note that when A is the image of a curve, N ∞ , is a pulled thread space as inProposition 4.1. Proof.
First observe that by the construction in Definition 5.3 there are maps(126) P j : M → N ∞ which are Riemannian isometries on regions which avoid A and map A to p .These define Riemannian isometries(127) P j : N j \ A (cid:48) j ˜ = M \ T δ j ( A ) → N ∞ \ T δ j ( p ) . In addition su ffi ciently small balls lying in these regions are isometric to convexballs in M .Observe also that for δ > ffi ciently small, the exponential map:(128) exp : { ( p , v ) : p ∈ A , v ∈ V p | v | < δ } → T δ ( A )is invertible where(129) V p = { v ∈ T p M : d M ( exp p ( tv ) , p ) = d M ( exp p ( tv ) , A ) } . Taking δ = δ A > A ), we canguarantee that ∀ v i ∈ V p , | v i | < δ A , t i ∈ (0 ,
1) we have(130) d M ( exp p ( t v ) , exp p ( t v )) ≤ d M ( exp p ( v ) , exp p ( v )) + | t − t | . This is not true unless A is a smooth embedded compact submanifold witheither no boundary or a smooth boundary.Define F j : N j → N ∞ as follows:(131) F j ( x ) = P j ( x ) ∀ x ∈ N j \ T δ j ( A (cid:48) j )and(132) F j ( x ) = p ∀ x ∈ A (cid:48) j . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 27
Between these two regions we take(133) F j ( x ) = f j ( P j ( x )) ∀ x ∈ T δ j ( A (cid:48) j ) \ A (cid:48) j where f j : N ∞ → N ∞ is a surjective map:(134) f j : Ann p ( δ j , δ j ) → B δ j ( p ) \ { p } which takes a point q to(135) f j ( q ) = γ q (cid:16) ( d N ∞ ( p , q ) − δ j ) /δ j (cid:17) where γ q is the unique minimal geodesic from γ q (0) = p to γ q (1) = q . Here weare assuming δ j < δ A . So(136) d N ∞ ( p , P j ( x )) = d M ( A , x )and(137) γ q ( t ) = P j ( exp q (cid:48) ( tv (cid:48) )) where P j ( exp q (cid:48) ( v (cid:48) )) = q . In particular for x ∈ ∂ T δ j ( A (cid:48) j ),(138) f j ( P j ( x )) = γ P j ( x ) ((2 δ j − δ j ) /δ j ) = γ P j ( x ) (1) = P j ( x )and for x ∈ ∂ A (cid:48) j ,(139) f j ( P j ( x )) = γ P j ( x ) (( δ j − δ j ) /δ j ) = γ P j ( x ) (0) = p so that F j is continuous.We claim Lip( F j ) = A (cid:48) j (140) Lip( F j ) ≤ T δ j ( A (cid:48) j ) \ A (cid:48) j (141) Lip( F j ) = N j \ T δ j ( A (cid:48) j ) . (142)Only the middle part is di ffi cult. By the definition of d N ∞ we have the followingtwo possibilitiesCase I: d N ∞ ( q , q ) = d M ( P − j ( q ) , P − j ( q ))(143) Case II: d N ∞ ( q , q ) = d M ( P − j ( q ) , A ) + d M ( P − j ( q ) , A ) . (144)In Case II we see that the minimal geodesic from q to q passes through p . Since f j ( q ) and f j ( q ) lie on this geodesic, we have(145) d N ∞ ( f j ( q ) , f j ( q )) ≤ d N ∞ ( q , q ) . In Case I we apply (130) with(146) t i = ( d M ( P − j ( q i ) , A ) − δ j ) /δ j because t i ∈ (0 ,
1) due to (141) so that by the reverse triangle inequality | t − t | = | d M ( P − j ( q ) , A ) − d M ( P − j ( q ) , A ) | /δ j (147) ≤ d M ( P − j ( q ) , q ) /δ j (148) ≤ d N ∞ ( q , q )(149) to see that d N ∞ ( f j ( q ) , f j ( q )) ≤ d M ( P − j ( f j ( q )) , P − j ( f j ( q )))(150) ≤ d M ( P − j ( q ) , P − j ( q )) + | t − t | by (130) , (151) ≤ d N ∞ ( q , q ) + | t − t | by Case I hypothesis,(152) ≤ d N ∞ ( q , q ) . (153)This gives our claim.We claim Lip( F j ) ≤ x , x ∈ N j , we have a minimizinggeodesic η : [0 , → N j such that η (0) = x and η (1) = x . Then(154) d N ∞ ( F j ( x ) , F j ( x )) ≤ L ( F j ◦ η ) . Since | ( F j ◦ η ) (cid:48) ( t ) | ≤ | η (cid:48) ( t ) | by our localized Lipschitz estimates and because thefunction F j is continuous, we are done. (cid:3) Constructing Almost Isometries.
See Section 2.2 for a review of the Gromov-Hausdor ff distance. Lemma 5.5.
Given N j as in Definition 5.3, the maps F j : N j → N ∞ defined in(131)-(133) in the proof of Lemma 5.4 are H j -almost isometries with lim j →∞ H j = . Thus (155) N j GH −→ N ∞ . Proof.
Before we begin the proof recall that(156) Diam( A (cid:48) j ) ≤ H j → ffi ces to show that F j are H j -almost isometries. To see this, examine x , y ∈ N j and join them by a minimizingcurve σ : [0 , → N j .If σ [0 , ⊂ N j \ A (cid:48) j , then by (131) we have(157) L ( σ ) = L ( F j ◦ σ )and so(158) d N j ( x , y ) ≥ d N ∞ ( F j ( x ) , F j ( y )) . Otherwise we have d N j ( x , y ) ≥ d N j ( x , A (cid:48) j ) + d N j ( y , A (cid:48) j ) T δ j ( A (cid:48) j ) to A (cid:48) j (159) = d N ∞ ( F j ( x ) , B δ j ( p )) + d N ∞ ( F j ( y ) , B δ j ( p ))(160) = d N ∞ ( F j ( x ) , p ) − δ j + d N ∞ ( F j ( y ) , p ) − δ j (161) ≥ d N ∞ ( F j ( x ) , F j ( y )) − δ j . (162)Next we join F j ( x ) to F j ( y ) by a minimizing curve γ . If γ [0 , ⊂ N ∞ \ B δ j ( p )then there is a curve η such that γ = F j ◦ η with η [0 , ⊂ N j \ A (cid:48) j and so by (131)(163) d N j ( x , y ) ≤ L ( η ) = L ( γ ) = d N ∞ ( F j ( x ) , F j ( y )) . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 29
Otherwise we have d N j ( x , y ) ≤ d N j ( x , A (cid:48) j ) + Diam( A (cid:48) j ) + d N j ( y , A (cid:48) j )(164) ≤ d N j ( x , A (cid:48) j ) + H j + d N j ( y , A (cid:48) j )(165) = d N ∞ ( F j ( x ) , B δ j ( p )) + d N ∞ ( F j ( y ) , B δ j ( p )) + H j (166) ≤ L ( γ ) + H j = d N ∞ ( F j ( x ) , F j ( y )) + H j . (167)Hence, F j is an H j isometry since 2 δ j < H j . (cid:3) Metric Measure Convergence.
Recall metric measure convergence as re-viewed in Section 2.3.
Lemma 5.6.
Given N j → N ∞ as in Lemma 5.4 endowed with the Hausdor ff mea-sures, then we have metric measure convergence if A has H -measure .Proof. Recall the maps F j : N j → N ∞ defined in (131)-(133) in the proof ofLemma 5.4. We need only show that for almost every p ∈ N ∞ and for almost every r < r p su ffi ciently small we have(168) H ( B ( p , r )) = lim j →∞ H ( B ( p j , r ))where F j ( p j ) = p and that for any sequence p j → p we have r su ffi ciently smallthat for all r < r (169) H ( B ( p , r )) = lim j →∞ H ( B ( p j , r )) . In fact take any p (cid:44) p in N ∞ and choose(170) r < r p < d N ∞ ( p , p ) / . Then for j large enough that δ j < r p we have(171) B ( p , r ) ∩ B ( p , δ j ) = ∅ . Thus(172) B ( p j , r ) ∩ A (cid:48) j = ∅ . Thus by (131), F j is an isometry from B ( p j , r ) ⊂ N j onto B ( p , r ) ⊂ N ∞ and so wehave(173) H ( B ( p , r )) = H ( B ( p j , r )) ∀ r < r p . Next we examine p . Observe that by (108)(174) H N ∞ ( B ( p , r )) = H M ( T r ( A )) − H M ( A ) = Vol M ( T r ( A ) \ A ) . For any p , j → p , we have by (125)(175) r j = d N j ( p , j , A (cid:48) j ) ≤ d N ∞ ( F j ( p , j ) , p ) → B ( p , j , r ) ⊂ T r + r j ( A (cid:48) j ) . So Vol N j ( B ( p , j , r )) ≤ Vol N j ( T r + r j ( A (cid:48) j ))(177) ≤ Vol N j ( T r + r j ( A (cid:48) j ) \ A (cid:48) j ) + Vol N j ( A (cid:48) j )(178) = Vol M (cid:16) T r + r j + δ j ( A ) \ T δ j ( A ) (cid:17) + Vol N j ( A (cid:48) j ) . (179)Thuslim sup j →∞ Vol N j ( B ( p , j , r )) ≤ Vol M ( T r ( A ) \ A ) + lim sup j →∞ Vol N j ( A (cid:48) j )(180) = H ( B ( p , r ))(181)since we claim that(182) lim j →∞ Vol N j ( A (cid:48) j ) = . This follows because (cid:15) j → − (cid:15) j ) Vol M ( A δ j ) ≤ Vol N j ( A (cid:48) j ) ≤ (1 + (cid:15) j ) Vol M ( A δ j ) . The assumption that H ( A ) = j su ffi ciently large(184) T r − H j − r j ( A (cid:48) j ) ⊂ B ( p , j , r ) . So Vol N j ( B ( p , j , r )) ≥ Vol N j ( T r − H j − r j ( A (cid:48) j ))(185) = Vol N j ( T r − H j − r j ( A (cid:48) j ) \ A (cid:48) j ) + Vol N j ( A (cid:48) j )(186) = Vol M (cid:16) T r − H j − r j + δ j ( A ) \ T δ j ( A ) (cid:17) + Vol N j ( A (cid:48) j ) . (187)Thus lim inf j →∞ Vol N j ( B ( p , j , r )) ≥ Vol M ( T r ( A ) \ A ) + lim inf j →∞ Vol N j ( A (cid:48) j )(188) = H ( B ( p , r )) , by (182)(189)which completes the proof. (cid:3) Intrinsic Flat Convergence.
For a review of intrinsic flat convergence seeSection 2.5.
Lemma 5.7.
Let N j GH −→ N ∞ be exactly as in Lemma 5.4 and Lemma 5.5 where weassume M is compact and we have a compact set, A ⊂ M \ ∂ M. Then there existsan integral current space N such that ¯ N is isometric to N ∞ and (190) N j F −→ N . and when A has Hausdor ff measure M ( N j ) → M ( N ) = H ( N ) . When A = C ([0 , then N = N ∞ . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 31
Proof.
By (123), we have uniformly bounded volume(192) Vol( N j ) ≤ M ) . Since ∂ N j = ∂ M , we have uniformly bounded boundary volume(193) Vol( ∂ N j ) = Vol( ∂ M ) . Combining this with Lemma 5.5 and Theorem 2.6, there exists an integral currentspace N possibly N = such that a subsequence(194) N j F −→ N . We claim that N (cid:44) . If not, then by the final line in Lemma 2.12, for anysequence p j ∈ N j and almost every r , S ( p j , r ) F −→ . However, taking p j and r such that(195) B ( p j , r ) ⊂ N j \ A (cid:48) j we know there is some p ∈ M with B ( p , r ) ⊂ N ∞ \ { p } that d F ( S ( p j , r ) , S ( p , r )) = p ∈ M , so S ( p j , r ) F −→ S ( p , r ) (cid:44) which is a contradiction.By Theorem 2.13, we know that after possibly taking a subsequence we obtaina limit map(196) F ∞ : N → N ∞ . We claim that F ∞ is distance preserving. Let p , q ∈ N . By Theorem 2.11, wehave p j , q j ∈ N j converging to p , q in the sense of Definition 2.9, i.e.(197) d N j ( p j , q j ) → d N ( p , q ) . Since the F j are (cid:15) j -almost isometries and (cid:15) j →
0, we have(198) d N ∞ ( F j ( p j ) , F j ( q j )) → d N ( p , q ) . By the definition of F ∞ we have F j ( p j ) → F ∞ ( p ) and F j ( q j ) → F ∞ ( q ). Thus(199) d N ∞ ( F ∞ ( p ) , F ∞ ( q )) = d N ( p , q ) . We claim that F ∞ maps onto at least N ∞ \ { p } . Let x ∈ N ∞ \ { p } . Since F j aresurjective, there exists x j ∈ N j such that F j ( x j ) = x . Since x (cid:44) p , we may define(200) r = min { d N ∞ ( x , p ) / , ConvexRad M ( x ) } where ConvexRad M ( x ) is the convexity radius about x viewed as a point in M .Then there exists j su ffi ciently large such that δ j < r so that(201) B ( x j , r ) ⊂ N j \ T δ j ( A (cid:48) j ) . Furthermore, these balls are isometric to the convex ball B ( x , r ) ⊂ M .So(202) d F ( S ( x j , r ) , ) = d F ( S ( x , r ) , ) > . Thus by Theorem 2.14 with h = d F ( S ( x , r ) , ), and N j F −→ N , a subsequence ofthe x j converges to x ∞ ∈ N . By the definition of F ∞ , we have F j ( x j ) → F ∞ ( x ∞ ) ∈ N ∞ . But since F j ( x j ) = x it follows that F ∞ ( x ∞ ) = x , hence F ∞ maps onto N ∞ \ p . Taking the metric completions of N and N ∞ \ { p } , we have an isometry(203) F ∞ : ¯ N → N ∞ . Since N j are Riemannian manifolds,(204) M ( [ N j ] ) = Vol( N j ) = H ( N j ) . By the lower semicontinuity of mass and the metric measure convergence of N j to N we know that(205) M ( [ N ∞ ] ) ≤ lim inf j →∞ M ( [ N j ] ) = H ( N ) . On the other hand by (29)(206) M ( [ N ∞ ] ) ≥ H ( N )because almost every tangent cone is Euclidean and it has integer weight every-where. Thus we have (191). In fact equality in these inequalities implies that N has weight one everywhere.Recall that the set of an integral current space only includes points of positivedensity. Since(207) lim inf r → Vol N ∞ ( B ( p , r )) r = lim inf r → Vol M ( T r ( A ) \ A ) r Thus N is isometric to N ∞ when this liminf is positive and N is isometric to N ∞ \{ p } when this liminf is 0. When A = C ([0 , r → Vol M ( T r ( A ) \ A ) r = lim inf r → π r L ( C ) r = + ∞ > . Thus N is isometric to N ∞ .Thus N does not depend on the subsequence in (194) and in fact the originalsequence (given a consistent orientation) converges in the intrinsic flat sense to N . (cid:3) The proof of Theorem 5.2.
Proof.
In Proposition 3.3 we show that given any (cid:15) j → n j → ∞ and δ j → δ j n j → h ( δ j ) n j → N j = ( M \ A δ j ) (cid:116) A (cid:48) δ j , satisfy:(210) (1 − (cid:15) ) Vol( A δ ) ≤ Vol( A (cid:48) δ ) ≤ Vol( A δ )(1 + (cid:15) )and(211) (1 − (cid:15) ) Vol( M ) ≤ Vol( N ) ≤ Vol( M )(1 + (cid:15) )and(212) Diam( A (cid:48) δ ) ≤ H ( δ ) = L ( C ) / n + ( n + h ( δ ) + (5 n + δ. EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 33 where(213) lim δ → H ( δ ) = K ∈ (0 , . Thus we have a sequence N j which is scrunching a set A = C ([0 , N j GH −→ N ∞ where N ∞ is the pulled string space. Lemma 5.6 implies we have metric measureto N ∞ convergence because A = C ([0 , H -measure 0.Lemma 5.7 implies that(215) N j F −→ N ∞ and(216) M ( N j ) → M ( N ∞ ) = H ( N ) , completing the proof of Theorem 5.2. (cid:3)
6. S ewing a S phere to O btain our L imit S pace Here we construct the specific example of a sequence of manifolds with positivescalar curvature that converges to a limit space which fails to have generalizednonnegative scalar curvature as discussed in the introduction. More specifically:
Example 6.1.
We define a sequence N j of manifolds with positive scalar curvatureconstructed from the standard S sewn along a closed geodesic C : [0 , → S with δ = δ j → as in Proposition 3.3. Then by Theorem 5.2 we have (217) N j mGH −→ N ∞ and N j F −→ N ∞ where N ∞ is the metric space created by taking the standard sphere and pullingthe geodesic to a point as in Proposition 4.1. By Lemma 6.3 below we see that atthe pulled point p ∈ N ∞ we have (3). Thus we have produces a sequence of threedimensional manifolds with positive scalar curvature converging to a limit spacewhich fails to satisfy generalized scalar curvature defined using limits of volumesof balls as in (1). Remark 6.2.
Note that with δ j → , the neck in the center of the tunnels has arotationally symmetric minimal surface whose area is ≤ πδ j which converges to . So this sequence, and in fact any sewn sequence created as in Definition 5.1,has MinA ( N j ) → . Lemma 6.3.
At the pulled point p ∈ N ∞ of Example 6.1 we have (218) lim r → (cid:32) Vol E ( B (0 , r )) − Vol N ∞ ( B ( p , r )) r Vol E ( B (0 , r )) (cid:33) = −∞ . Proof.
First, observe thatVol N ∞ ( B ( p , r )) = H N ∞ ( B ( p , r ))(219) = H N ∞ ( B ( p , r ) \ { p } )(220) = H S ( T r ( C ([0 , . (221)Since C ([0 , π in a three dimensional sphere, wehave(222) lim r → H S ( T r ( C ([0 , π ( π r ) = . Thus(223) lim r → Vol E ( B (0 , r )) − Vol N ∞ ( B ( p , r )) r Vol E ( B (0 , r )) = lim r → (4 / π r − π ( π r )(4 / π r = −∞ as claimed. (cid:3)
7. A ppendix : S hort tunnels with P ositive S calar C urvatureby J orge B asilio and J´ ozef D odziuk There is a deep connection between the geometry of Riemannian manifolds M n with positive scalar curvature and surgery theory. The subject began with the sur-prising discovery by Gromov and Lawson [GL80b] (for n ≥
3) and Schoen and Yau[SY79a] that a manifold obtained via a surgery of codimension 3 from a manifold M n with a metric of positive scalar curvature may also be given a metric with pos-itive scalar curvature. The key to the tunnel construction of [GL80b] is defining acurve γ which begins along the vertical axis then bends upwards as it moves to theright and ends with a horizontal line segment, cf. Figure 6 below. The tunnel thenis the surface of revolution determined by γ . We note that the “bending argument”has attracted some attention (See [RS01]).As the goals of the surgery theory were topological in nature Gromov and Law-son did not estimate with diameters or volumes of these tunnels. Indeed, the tun-nels they constructed may be thin but long (See [GL80a]). To build sewn manifoldswe need tunnels with diameters shrinking to zero as the size of the original ballsdecreases to zero (see (7), (8) (9)). Therefore, we prove Lemma 2.1 to obtain arefinement of the Gromov and Lawson construction showing the existence of tiny(in sense of (10)) and arbitrarily short tunnels with a metric of positive scalar cur-vature. Proof of Lemma 2.1.
To aid the reader, we provide a summary of our proof andintroduce additional notation.7.1.
Outline of Proof of Lemma 2.1.
To aid the reader, we provide a summary ofour proof and introduce additional notation.
Step 1: Setup and notation.
Let (cid:15) > < δ < δ/ B = B ( p , δ/ ⊂ M has constant sectional curvature K > EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 35
This is also true for B ( p , δ ) ⊂ B for 0 < δ < δ/ p .Thus, B ( p , δ ) is a hypersurface of revolution U (cid:48) γ with the induced metric in R determined by revolving a segment of the circle γ in the ( x , x )-plane about the x -axis. We set things up so that the vertical x -axis corresponds to boundarypoints of B ( p , δ ). We then proceed as Gromov and Lawson to deform γ awayfrom vertical axis bending it upwards as we move to the right and ending with anarbitrarily short horizontal line segment. We call this curve γ , cf. Figure 6. Thecurve γ begins exactly as γ so that we may attach the corresponding hypersurfaceonto the larger B ( p , δ/
2) in a natural way. We do exactly the same for B ⊂ M and identify the two hypersurfaces along their common boundary, i.e the “tinyneck,” forming 2 U (cid:48) γ = U (cid:48) γ (cid:116) U (cid:48) γ . We then define the tunnel U = U δ by(224) U = U δ = (( B ( p , δ/ \ B ( p , δ )) (cid:116) (2 U (cid:48) δ ,γ ) (cid:116) (( B ( p , δ/ \ B ( p , δ )) , where 0 < δ < δ/ U (cid:48) γ = U (cid:48) δ ,γ is a modified Gromov-Lawson tunnel, seeFigure 1.The boundary of 2 U (cid:48) γ is isometric to a collar of B ( p , δ ) (cid:116) B ( p , δ ) so we maysmoothly attach it to form (224). Step 2: Construction of the curve γ , Part 1: C . In this step, we construct a C ,and piecewise C ∞ , curve γ . The construction is based on the bending argumentof Gromov and Lawson and uses the fundamental theorem of plane curves i.e. thefact that a smooth curve parametrized by arclength is uniquely determined by itscurvature, the initial point and the initial tangent vector. Care must be taken toensure that the induced metric on U (cid:48) γ maintains positive scalar curvature and thatthe legth of γ is controlled to yield diameter and volume estimates of Lemma 2.1.This step is quite technical and forms the heart of the proof. Step 3: Construction of the curve γ , Part 2: from C to C ∞ . In this step we show how to modify the curve constructed in Step 2 to obtain asmooth curve ¯ γ while maintaining all the required features. The modification iselementary and, once it is completed, we rename ¯ γ back to γ . Step 4: Diameter estimates (7), (9) and volume estimates (10), (11).
This is very straightforward since the previous steps give an estimate of the lengthof the tunnel.We remark here that the choice of δ is used only to insure that the tunnel U (cid:48) (see Figure 1) has su ffi ciently small volume.7.2. Step 1 of the Proof.
We now set-up our notation further, describe U explicitlyin terms of a special curve γ , and state the important curvature formulas needed inlater steps. The construction of γ is done in the next two sub-sections (Steps 2 and3). As mentioned in subsection 7.1, because we assume that B and B have con-stant sectional curvature K we may work directly in Euclidean space R with co-ordinates ( x , x , x , x ) and its standard metric. Let γ ( s ) be a curve in the ( x , x )-plane, parametrized by arc-length, written as γ ( s ) = ( x ( s ) , x ( s )). This curvespecifies a hypersurface in R (by rotating γ about the x -axis),(225) U (cid:48) = U (cid:48) γ = { ( x , x , x , x ∈ R | x = x ( s ) , x + x + x = x ( s ) } , which we endow with the induced metric. Our curve γ will always lie in the firstquadrant of ( x , x )-plane and will be parametrized so that x ( s ) will be increasing.We denote by θ ( s ) the angle between the horizontal direction and the upward nor-mal vector, and by ϕ ( s ) the angle between the horizontal direction and the tangentvector to γ . F igure
6. The curve γ .We remark that the two angle functions are related by(226) θ ( s ) = ϕ ( s ) + π , see Figure 6. In particular, ϕ ∈ ( − π/ , k ( s ) the geodesic curvature of γ . It is a signed quantity so that γ bendsaway from the horizontal axis if k ( s ) > x -axis when k ( s ) < γ ( s ) = ( c , d ) and ϕ = ϕ ( s ) then (cf. Theorem 6.7, [Gra98]) the function k ( s )determines γ by the formulae(227) ϕ ( s ) = ϕ + (cid:90) ss k ( u ) du and(228) γ ( s ) = (cid:32) c + (cid:90) ss cos( ϕ ( u )) du , d + (cid:90) ss sin( ϕ ( u )) du (cid:33) . Our aim is to define a function k ( s ) so that the resulting threefold of revolution U (cid:48) has positive scalar curvature. The formula on page 226 of [GL80b] for n = U (cid:48) ( s ) = θ ( s ) x ( s ) (cid:34) sin θ ( s ) x ( s ) − k ( s ) (cid:35) EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 37 where Scal U (cid:48) ( s ) is the scalar curvature of the induced metric on U (cid:48) and k is thegeodesic curvature of γ . In particular, the formula holds if γ is the intersection ofthe 3-sphere around the origin with the ( x , x )-plane in which case k is a negativeconstant.We begin defining our curve γ ( s ) so that γ (0) corresponds to a point on ∂ B ( p , δ )and γ ( s ), for small values of s ∈ [0 , s ], parametrizes the intersection of B ( p , δ )with the ( x , x )-plane. In particular, for small s , k ( s ) ≡ − √ K . We choose s = δ / k ( s ) to a suit-able step function on a longer interval [0 , L ] so that the resulting curve γ ( s ) has thefollowing properties.(I) The graph of γ lies strictly in the first quadrant, beginning at p I = γ (0) = (0 , cos( − π/ + δ ) / √ K ) and ending at p F = γ ( L ) with x ( L ) > x ( L ) > L is the length of the curve. Moreover, a point of γ moves to the rightwhen s increases.(II) Let θ ( s ) be the angle between the upward pointing normal to γ and the x -axis. The curve γ ends at p F with θ ( L ) = π/ θ = π/ L (cid:48) , L ] (where L (cid:48) < L ).(III) The curve γ has constant curvature − √ K near 0 so that the boundary of U has a neighborhood that is isometric to a collar of B ∪ B .(IV) The curvature function k ( s ) satisfies(230) k ( s ) < sin( θ ( s ))2 x ( s ) s ∈ [0 , L ] , so that the expression on the right-hand side of (229) is positive for all s ∈ [0 , L ]. We remark here that in certain stages of the construction k ( s )will have discontinuities so that Scal U (cid:48) ( s ) is not defined but this will causeno di ffi culties.(V) The length of γ , L , is O ( δ ).Due to properties (I) and (II) of γ above, we may smoothly attach two copies of U (cid:48) along their common boundary at s = L to define 2 U (cid:48) = U (cid:48) γ (cid:116) U (cid:48) γ and then, usingproperty (III), attach 2 U (cid:48) to form U as in (224).In the next step, we construct a piecewise C curve γ in the ( x , x )-plane whichsatisfies properties (I) through (V). Then, in Step 3, we modify the constructiononce more to produce a smooth curve, ¯ γ , with these same properties.7.3. Step 2 of the Proof: Construction of γ , Part 1: C . As above, let s = δ / q = ( a , b ) be the coordinates of the point γ ( s ) that is already defined. Bychoosing δ su ffi ciently small we can assume that the tangent vector to γ at s = s is nearly vertical and is pointing downward at s = s . We also have k ( s ) ≡ − √ K on [0 , s ].We will use a finite induction to define a sequence of extensions of γ over in-tervals [ s i , s i + ], with s i < s i + for a finite number of steps 0 ≤ i ≤ n , where n = n ( δ ) is the number of steps required such that properties (I), (III), (IV), and(V) all hold at each extension. We denote by ( a i , b i ) the coordinates of the point γ ( s i ) for 0 ≤ i ≤ n .Let us first choose the curvature function k ( s ) of γ ( s ) on the first extended in-terval [ s , s ]. Observe that equation (230) limits the amount of positive curvatureallowed for k ( s ). In fact, we choose k ( s ) to be the constant k > s , s ] based only the initial data at s (231) k = sin( θ ( s ))4 b > , where θ ( s ) = π + ϕ ( s ) = δ − √ K s > b = x ( s ). Note that constantpositive curvature means that γ ( s ) moves along the arc of a circle of curvature1 / √ k bending away from the origin.We verify that property (IV) holds with our choice of k in (231). From (227),we see that ϕ ( s ) is an increasing function with range in the interval ( − π/ , θ ( s ) is also increasing by (226). Moreover, from (227) and (228), we see that the x -coordinate function is decreasing on the interval ( s , s ) since x (cid:48) ( s ) = sin( ϕ ( s )) <
0. Thus, the expression on the right-hand side of (230), sin( θ ( s )) / (2 x ( s )), is anincreasing function on ( s , s ) so that(232) sin( θ ( s ))2 x ( s ) ≤ sin( θ ( s ))2 x ( s ) s ∈ [ s , s ] . Since k ( s ) ≡ k is constant it follows that the property (IV) holds for s ∈ [ s , s ].Next, we choose the length of the extension ∆ s = s − s , so that properties (I)and (V) hold. This is achieved by setting(233) ∆ s = b > x ( s ) is increasing since x (cid:48) ( s ) = cos( ϕ ( s )) > ϕ ∈ ( − π/ , b < δ since b is the vertical distance of γ ( s ) to the x -axis which is less than the distancealong the sphere.Of course, we do not achieve a final angle of π/ s and gainonly a small but definite increase in the angle. The change in angle of the normalwith the x -axis is ∆ θ = θ ( s ) − θ ( s ) = (cid:90) s s k ( s ) ds = k · ∆ s = sin( θ ( s ))8 > γ extended over the first interval [ s , s ], we now inductively define furtherextensions. Assume that ∆ s j , s j and k j have been chosen for j = , , . . . , ( i − γ extended on the intervals [ s j , s j + ], we then define(235) ∆ s i = b i − , s i = s i − + ∆ s i and k i = sin( θ ( s i − ))4 b i − , EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 39 where γ ( s i ) = ( a i , b i ). In what follows we will also write θ j and ϕ j for θ ( s j ) and ϕ ( s j ) respectively. We remark that b i + < b i by (228) since the angle ϕ is negativeand that k i + > k i since the ratio sin( θ ( s )) x ( s ) is increasing. Observe that properties (I),(IV), and (V) of γ hold on [ s i − , s i ] for all i by our choices in (235) by argumentsanalogous to those given for the first extension of γ on [ s , s ].We observe that we gain a definite amount of angle θ with each extension since,by (235), for each j ∈ { , , . . . , i } , ∆ θ j = θ ( s j ) − θ ( s j − ) = (cid:90) s i s j − k ( s ) ds = k j · ∆ s j = sin( θ ( s j − ))8 ≥ sin( θ ( s ))8 , (236)because θ ( s j − ) ≥ θ ( s ) and the the values of θ are in the range (0 , π/
2) so thatthe sine is an increasing function. We stop the construction when θ ( s ) reaches thevalue π/
2. Thus the total change in the angle θ over the interval [0 , s i ] is boundedfrom below by(237) ∆ θ = i (cid:88) j = ∆ θ j ≥ i · sin( θ )8 . To prove property (V), that the length of γ is on the order of δ , we need thesequence of b i ’s to be summable and will want to compare it to the geometricprogression. The di ffi culty here is that, since our curve is bending more and moreupwards, the ratios b i / b i − increase. For this reason we stop our induction when θ reaches the value of π/
4. It will turn out that once this value is reached, we cancomplete the construction of k ( s ) by a single extension albeit with ∆ s not given by(235).Thus, define n = n ( δ ) to be the first positive integer with(238) π ≤ θ n which exists by (237). Moreover, if θ n > π/ s n to be the exact valuein ( s n − , ∞ ) such that θ ( s n ) = π/
4. Thus, for the modified value of s n (239) θ n = θ ( s n ) = π . The following Lemma gives the desired comparison.
Lemma 7.1.
There exists a universal constant C ∈ (0 , , independent of δ andK, such that for all i ≤ n b i ≤ C · b i − , where n = n ( δ ) is as above. The Lemma, to be proven shortly below, implies that the length of the curve γ on the entire interval [0 , s n ] is no larger than a constant (independent of δ ) times δ . Namely,(240) L ( γ ([0 , s n ])) = s n = n (cid:88) j = ∆ s j . Thus, from (235) and Lemma (7.1), we have(241) n (cid:88) j = ∆ s j = n (cid:88) j = b j − ≤ b n − (cid:88) j = C j ≤ C δ by the lemma and (234). So, L ( γ ([0 , s n ])) ≤ C (cid:48) b with C (cid:48) = − C which is inde-pendent of δ since C is. This proves that L ( γ ([0 , s n ])) = O ( δ ). Proof of Lemma 7.1.
Let 1 ≤ i ≤ n . We compute explicitely using (227), (228) and(235),(242) ϕ ( s i ) = ϕ ( s i − ) + k i · ∆ s i = ϕ ( s i − ) + sin( θ i − )8and b i = x ( s i ) = b i − + (cid:90) s i s i − sin( ϕ ( s i − ) + k i ( u − s i − )) du = b i − − k i (cos( ϕ ( s i )) − cos( ϕ ( s i − ))) = b i − − b i − sin( θ ( s i − )) (cid:32) cos (cid:32) ϕ ( s i − ) + sin( θ i − )8 (cid:33) − cos( ϕ ( s i − )) (cid:33) . Thus, b i b i − = − θ ( s i − )) (cid:32) cos (cid:32) ϕ ( s i − ) + sin( θ i − )8 (cid:33) − cos( ϕ ( s i − )) (cid:33) . Therefore, by the Mean Value Theorem, there exists µ i ∈ ( ϕ ( s i − ) , ϕ ( s i − ) + sin( θ ( s i − )) / b i b i − = − θ ( s i − )) ( − sin( µ i )) · sin( θ ( s i − ))8 = + sin( µ i )2 . To complete the proof of the claim, we seek a constant 0 < C <
1, independent of δ , such that 1 + sin( µ i )2 < C < . (243)Recall that the angle function ϕ takes negative values throughout.We claim that the choice(244) C = +
14 sin (cid:32) − π + cos( − π )8 (cid:33) ≈ . EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 41
This follows from the fact that the sine is an increasing function on the interval( ϕ ( s i − ) , ϕ ( s i − ) + sin( θ ( s i − )) /
8) and the fact that both the angles ϕ i and θ i areincreasing, so 1 + sin( µ i )2 ≤ +
12 sin (cid:32) ϕ ( s i − ) + sin( θ ( s i − ))8 (cid:33) ≤ +
12 sin (cid:32) ϕ ( s n ) + cos( ϕ ( s n ))8 (cid:33) . By our choice of s n , θ ( s n ) = π/ ϕ ( s n ) = − π/ + sin( µ i )2 ≤ +
12 sin − π + cos (cid:16) − π (cid:17) < +
14 sin − π + cos (cid:16) − π (cid:17) = C < . This finishes the proof of the Lemma. (cid:3)
At this stage of the construction, γ has angle θ = π/ s n . Wemake one additional extension of our step function.We now define s n + > s n and k n + > ϕ ( s ) in [ s n , s n + ] will be given by(245) ϕ ( s ) = ϕ n + (cid:90) ss n k ( u ) du = ϕ n + k n + ( s − s n ) . Let s n + be determined by k n + as the first value such that ϕ ( s n + ) = θ ( s n + ) = π/ = ϕ ( s n + ) = ϕ n + k n + ( s n + − s n )so that(247) s n + = s n − ϕ n k n + . We require in addition that b ( s n + ) > γ remains above the x -axis). Using(247) and (228), we obtain b ( s n + ) = b n + (cid:90) s n + s n sin( ϕ ( s )) ds = b n − cos( ϕ ( s n + )) − cos( ϕ ( s n )) k n + = b n − − cos( ϕ ( s n )) k n + (248)so that b ( s n + ) > b n − − cos( ϕ ( s n )) k n + > k n + · b n > − cos( ϕ ( s n )) . On the other hand, k n + has to be bounded from above in order to guarantee(230). Therefore, we require that k n + < sin( θ ( s n ))2 b n , or(250) k n + · b n < sin( θ ( s n ))2 . Combining (249) and (250) gives conditions for k n + (251) 1 − cos( ϕ ( s n )) < k n + · b n < sin( θ ( s n ))2 . Since sin( θ ( s )) = cos( ϕ ( s )), (251) is equivalent to(252) 1 − cos( ϕ ( s n )) < k n + · b n < cos( ϕ ( s n ))2 . Now, recall that s n was chosen in (239) so that ϕ ( s n ) = − π/ − cos( ϕ ( s n )) = − √ < cos( ϕ ( s n ))2 = √ . Now, choose arbitrarily any α , satisfying(253) 2 − √ < α < √ k n + by(254) k n + = α/ b n . With this choice (252), and therefore, (249) and (250) hold.F igure
7. Graph of the curvature, k ( s ), with “full bend” as a step function.To ensure property (II), we choose L > s n + so that L − s n + is arbitrarily small.We extend γ to the interval [ s n + , L ] where γ is a straight horizontal line on [ s n + , L ]by choosing k ( s ) = O ( γ ) we observe that(255) s n + = s n − ϕ n / k n + = s n + π α b n ≤ s n + π α b = O ( δ ) EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 43 by (234), (241) and (255).We note that the choice of L is arbitrary. It will be made explicit in the next stepwhen we construct the curve ¯ γ , the C ∞ version of γ .This completes the construction of the continuously di ff erentiable curve γ de-fined on the interval [0 , L ] satisfying properties (I) through (V).7.4. Step 3 of the Proof: Construction of γ , Part 2: from C to C ∞ . In thissection, barred quantities will refer to the C ∞ curve ¯ γ ( s ) to be constructed in thisstep and all the other quantities related to the construction (for example, ¯ θ , ¯ ϕ , ¯ k ( s ),etc.). Unbarred quantities will refer to the C curve constructed in the previousstep.The general plan is to replace k ( s ) as chosen in Step 2 with a smooth version¯ k ( s ) as depicted in Figure 8, which will then define ¯ γ by the formulae (227) and(228). Set k = − K / and modify k ( s ) on [ s i , s i + ] for i = , , , . . . , n so thatthe graph of ¯ k ( s ) will connect to the constant function equal to k i smoothly at s i , will rise steeply to the value k i + in a very short interval [ s i , s i + α ] and willconnect smoothly with constant function equal to k i + in [ s i + α, s i + ]. For each i = , , , . . . n , ¯ k | [ s i , s i + ] can be constructed as follows. Choose and fix a C ∞ function g ( s ) which is identically 0 for s <
0, identically 1 for s >
1, and strictly increasingon [0 , k | [ s i , s i + ] is constructed by appropriate rescaling and translationsof the graph of g ( s ) in both vertical and horizontal directions. The values of k i and k i + determine the transformations along the vertical axis but rescaling of theindependent variable remains a free parameter α to be set su ffi ciently small later.We will use the same value of α for every i = , , . . . n .F igure
8. Graph of the smooth curvature ¯ k ( s ) with “full bend.”Since ∆ ¯ θ = (cid:90) s n + ¯ k ds ≤ (cid:90) s n + k ds = ∆ θ, we loose a small amount of ”bend” so that ¯ θ ( s n + ) < π by a very small amountcontrolled by α . We compensate for this by one final extension of ¯ k to an interval [ s n + , L ] with L = s n + + β . We choose ¯ k so that it connects smoothly with k n + at s n + , drops smoothly to zero over [ s n + , s n + + β ] and continues identically zero on[ s n + + β, s n + + β ]. β and ¯ k are chosen so that (cid:90) s n + + β s n + ¯ k ( s ) ds = π − ¯ θ ( s n + ) . This ensures that ¯ θ = π in the interval [ s n + + β, s n + + β ]. This final extension isconstructed as the preceding ones except that we have to use the reflection s (cid:55)→ − s before rescaling and translating the original fuction g . We note that β = O ( α ) isdetermined by the choice of α and the requirement that ¯ θ ( L ) = π . We also observethat as α tends to zero, the functions ¯ ϕ , ¯ θ , ¯ x , and ¯ x will converge uniformly on[0 , L ] to ϕ , θ , x , and x respectively as follows from (227) and (228).We now check that the properties (I) through (V) on page 37 hold for the curve¯ γ for su ffi ciently small choice of α . Only (IV) and (V) need a verification. (V)follows since L = s n + + β = O ( δ ) + O ( α ). To prove (IV) we use the uniformconvergence on [0 , s n + ] as α approaches 0 of sin ¯ θ ( s )2 ¯ x ( s ) to sin θ ( s )2 x ( s ) . More precisely, on[ s i , s i + ], sin ¯ θ ( s )2 ¯ x ( s ) − ¯ k ( s ) = (cid:32) sin ¯ θ ( s )2 ¯ x ( s ) − k i + (cid:33) + (cid:16) k i + − ¯ k ( s ) (cid:17) . For su ffi ciently small α , the first term on the right becomes positive by the property(IV) for the curve γ while the second term is nonnegative by construction (cf.Figure 7). Finally, in the last interval [ s n + , L ] the ratio sin ¯ θ ( s )2 ¯ x ( s ) is nondecreasing sothat sin ¯ θ ( s )2 ¯ x ( s ) ≥ sin ¯ θ ( s n + )2 ¯ x ( s n + ) > k n + since the last inequality was verified for s = s n + already. Property (IV) followssince k n + > ¯ k ( s ) in [ s n + , L ]. This finishes the construction of ¯ γ .7.5. Step 4 of the Proof: Diameter and volume estimates of Lemma 2.1.
Giventhe definition of U in (224), the diameter of U is estimated byDiam( U ) ≤ πδ + δ + L = O ( δ ) + O ( δ ) = O ( δ ) . To estimate the volume of U (cid:48) , note that the intersection of U (cid:48) with the hyperplane x = x ( s ) = c for 0 < s < L is a sphere of two dimensions and of radius x ( s ) < δ .It follows by Fubini’s theorem that Vol( U (cid:48) ) = O ( δ ). To prove (10) recall that U isobtained from the union of two disjoint balls of radius δ by removing balls of radius δ and attaching U (cid:48) along the common boundary (cf. Figure 1). Since the volumesof the removed balls and of the added tunnel are O ( δ ), the estimate (10) followsby choosing δ su ffi ciently small depending on (cid:15) . The estimate (11) is proved inthe same way. The proof of Lemma 2.1 is now complete. (cid:3) EWING RIEMANNIAN MANIFOLDS WITH POSITIVE SCALAR CURVATURE 45 R eferences [AGS14] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´e. Metric measure spaces with Rie-mannian Ricci curvature bounded from below. Duke Math. J. , 163(7):1405–1490, 2014.[AK00] Luigi Ambrosio and Bernd Kirchheim. Currents in metric spaces.
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