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We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences o...
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... different tech- niques, Gromov-Lawson and Schoen-Yau described how to construct tunnels dif- feomorphic to S 2 × [0, 1] with metric tensors of positive scalar curvature that can be glued smoothly into three dimensional spheres of constant sectional curvature [GL80b] [SY79a]. See Figure 1. These tunnels are the first crucial piece for our construction. ...
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... completes the construction of the continuously differentiable curve γ de- fined on the interval [0, L] satisfying properties (I) through (VI). See Figure 13. ...
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... general plan is to replace k(s) as chosen in Step 2 with a smooth version, ¯ k(s), as depicted in Figure 14 and whose formula is given below in (268), (273), We'll use a smooth functions that increase from k i to k i+1 to over an arbitrarily short interval to construct ¯ γ. The building block for these functions is the smooth function h : R → [0, 1] defined by Let η i > 0 represent quantities which can be chosen arbitrarily small satisfying s i + η i < s i+1 for i = 0, 1, . . . ...
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... define ¯ k(s) as the function that smoothly increases from k i at s i to k i+1 at s i + η i and then continues with this constant value until s i+1 . More precisely, if we set 1, 2, . . . , n describes the smooth curvature function in Figure 14. ...
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... precisely, if we set 1, 2, . . . , n describes the smooth curvature function in Figure 14. Of course, we choose the same initial conditions as in k(s) in the previous step, namely, the choice of the constants in (237). ...
Citations
... Thirdly, we can use the volume-limit notion to characterize scalar curvature or scalar curvature lower bounds. This notion is used by Basilio, Dodziuk, and Sormani [2], Basilio and Sormani [3], and Kazaras and Xu [10] to show that the limit space has negative scalar curvature at some point. ...
... In [15], the author worked with Jiewon Park and Changliang Wang to confirm Gromov's conjecture under the additional MinA lower bound for sequences of rotationally symmetric Riemannian manifolds. In [17], the author worked with Changliang Wang to explore Gromov's conjecture under the additional MinA lower bound for sequences of warped product S 2 × S 1 , we prove that the sequence of Riemannian metric converges to a limit Riemannian metric that is W 1,p for all p ∈ [1,2). We also prove that the limit space has non-negative distributional scalar curvature as defined in [12]. ...
... After possibly passing to a subsequence, the sequence of Riemannian metric tensor g j converges in L q norm to a limit metric g ∞ for all q ∈ [1, ∞). The limit metric g ∞ is in W 1,p for all [1,2). The Riemannian manifold (S 2 × S 1 , g ∞ ) has a W 1,p metric tensor with non-negative distributional scalar curvature as defined in [12]. ...
In this article, we extend the example constructed in the paper by Sormani-Tian-Wang to build new examples that satisfy the assumptions of the conjecture by Gromov. Each of these new examples of sequence converges to a limit space with infinitely many poles in $\mathbb{S}^2$. These examples can be used to test various notions of weak scalar curvature.
... In [7,8], Gromov conjectured that a sequence of Riemannian manifolds with nonnegative scalar curvature, Scalar ≥ 0, should have a subsequence which converges in some weak sense to a limit space with some generalized notion of "nonnegative scalar curvature". In light of the examples constructed by Basilio, Dodziuk, and Sormani in [2], the MinA condition in (2) below was added to prevent collapsing happening, and the conjecture was made more precise at an IAS Emerging Topics Workshop co-organized by Gromov and Sormani as follows [18]: ...
... In [7,8], Gromov conjectured that a sequence of Riemannian manifolds with nonnegative scalar curvature, Scalar ≥ 0, should have a subsequence which converges in some weak sense to a limit space with some generalized notion of "nonnegative scalar curvature". In light of the examples constructed by Basilio, Dodziuk, and Sormani in [2], the MinA condition in (2) below was added to prevent collapsing happening, and the conjecture was made more precise at an IAS Emerging Topics Workshop co-organized by Gromov and Sormani as follows [18]: ...
... for all p ∈ [1,2) in the sense of Definition 5.3. ...
Gromov and Sormani conjectured that a sequence of three dimensional Riemannian manifolds with nonnegative scalar curvature and some additional uniform geometric bounds should have a subsequence which converges in some sense to a limit space with some generalized notion of nonnegative scalar curvature. In this paper, we study the pre-compactness of a sequence of three dimensional warped product manifolds with warped circles over standard \({{\mathbb {S}}}^2\) that have nonnegative scalar curvature, a uniform upper bound on the volume, and a positive uniform lower bound on the \({\text {MinA}}\), which is the minimum area of closed minimal surfaces in the manifold. We prove that such a sequence has a subsequence converging to a \(W^{1, p}\) Riemannian metric for all \(p<2\), and that the limit metric has nonnegative scalar curvature in the distributional sense as defined by Lee–LeFloch.
... Lee-Naber-Neumayer [LNN20] discovered analogous drawstring phenomena in dimensions 4 and above (see also Lee-Topping [LT22]). Prior to [LNN20], Basilio-Dodziuk-Sormani [BDS18] worked in dimension 3 and developed a different construction called sewing (see also [BS21]). This construction led them to the first examples of positive scalar curvature manifolds that converge to pulled string spaces. ...
... We refer to (Y, d Y ) as created from (X, d X ) by pulling σ to a point. In the case where (X, d X ) carries an integral current structure, such as the case of a Riemannian manifold, an associated structure is induced on the pulled-string space, see [BDS18] for details. The next task is to articulate a principle that allows us to estimate the intrinsic flat distance between our constructions and pulled-string spaces. ...
... Definition 2.1. [BDS18,BS21] Let (M, g) be an oriented closed Riemannian manifold with a given compactly embedded curve σ ⊂ M . A sequence of oriented closed Riemannian manifolds ...
In this paper, we observe new phenomena related to the structure of 3-manifolds satisfying lower scalar curvature bounds. We construct warped-product manifolds of almost nonnegative scalar curvature that converge to pulled string spaces in the Sormani-Wenger intrinsic flat topology. These examples extend the results of Lee-Naber-Neumayer \cite{LNN} to the case of dimension $3$. As a consequence, we produce the first counterexample to a conjecture of Sormani \cite{SormaniConj} on the stability of the Geroch Conjecture. Our example tests the appropriate hypothesis for a related conjecture of Gromov. On the other hand, we demonstrate a $W^{1,p}$-stability statement ($1\leq p<2$) for the Geroch Conjecture in the class of warped products.
... In [8] and [7], Gromov conjectured that a sequence of Riemannian manifolds with nonnegative scalar curvature, Scalar ≥ 0, should have a subsequence which converges in some weak sense to a limit space with some generalized notion of "nonnegative scalar curvature". In light of the examples constructed by Basilio, Dodziuk, and Sormani in [2], the MinA condition in (2) below was added to prevent collapsing happening, and the conjecture was made more precise at an IAS Emerging Topics Workshop co-organized by Gromov and Sormani as follows [15]: Then there exists a subsequence which is still denoted as {M j } ∞ j=1 that converges in the volume preserving intrinsic flat sense to a three dimensional rectifiable limit space M ∞ . Furthermore, M ∞ is a connected geodesic metric space, that has Euclidean tangent cones almost everywhere, and has nonnegative generalized scalar curvature. ...
... In [8] and [7], Gromov conjectured that a sequence of Riemannian manifolds with nonnegative scalar curvature, Scalar ≥ 0, should have a subsequence which converges in some weak sense to a limit space with some generalized notion of "nonnegative scalar curvature". In light of the examples constructed by Basilio, Dodziuk, and Sormani in [2], the MinA condition in (2) below was added to prevent collapsing happening, and the conjecture was made more precise at an IAS Emerging Topics Workshop co-organized by Gromov and Sormani as follows [15]: Then there exists a subsequence which is still denoted as {M j } ∞ j=1 that converges in the volume preserving intrinsic flat sense to a three dimensional rectifiable limit space M ∞ . Furthermore, M ∞ is a connected geodesic metric space, that has Euclidean tangent cones almost everywhere, and has nonnegative generalized scalar curvature. ...
Gromov and Sormani conjectured that a sequence of three dimensional Riemannian manifolds with nonnegative scalar curvature and some additional uniform geometric bounds should have a subsequence which converges in some sense to a limit space with generalized notion of nonnegative scalar curvature. In this paper, we study the pre-compactness of a sequence of three dimensional warped product manifolds with warped circles over standard $\mathbb{S}^2$ that have nonnegative scalar curvature, a uniform upper bound on the volume, and a positive uniform lower bound on the MinA, which is the minimum area of closed minimal surfaces in the manifold. We prove that such a sequence has a subsequence converging to a $W^{1, p}$ Riemannian metric for all $p<2$, and that the limit metric has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch.
... The MinA condition in (2) can be viewed as a noncollapsing condition which prevents counter examples like sequences of round spheres rescaled to a point. Note that in a warped spheres with (3) g j = dr 2 + h j (r) 2 g S 2 where h j : [0, D j ] → (0, ∞) any level set r −1 (r 0 ) where h j (r 0 ) = 0 is a minimal surface of area 4πh j (r 0 ) 2 . In joint work with Jiewon Park, the second and third authors have proven Conjecture 1.1 for sequences of warped spheres with metric tensors of the form (3) [31]. ...
... The MinA condition in (2) was added to the conjecture in light of sequences of M 3 j satisfying the other three hypotheses (1) of this conjecture constructed by Basilio, Dodziuk, and the first author whose limit spaces do not satisfy the properties of spaces with a natural notion of generalized nonnegative scalar curvature [3]. These sequences have increasingly many increasingly tiny tunnels. ...
In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the $W^{1,p}$ sense for $p<2$ to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov-Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.
... Sormani proposed the MinA condition in [Sor17] to prevent bad limiting behavior, such as bubbling and pinching, along the sequence. The motivation for such a condition comes from the sewing construction of Basilio, Dodziuk, and Sormani [BDS18]. This construction shows the existence of a sequence of manifolds with positive scalar curvature, which has an F-limit that does not have positive scalar curvature in some generalized sense. ...
... Originally, the construction in [GL80] was used to make tunnels of positive scalar curvature to show, for example, that the connected sum of two manifolds with positive scalar curvature carries a metric of positive scalar curvature. For manifolds with constant positive sectional curvature, Dodziuk, Basilio, and Sormani in [BDS18] refined the construction to give control over the volume and diameter of the tunnel while maintaining positive scalar curvature. Dodziuk in [Dod20] further refined the construction by replacing the positive sectional curvature condition with positive scalar curvature. ...
... The new tunnel construction allows us to extend the sewing construction in [BDS18] and [BS21] to a more general setting. Basilio, Dodziuk, and Sormani [BDS18] used sewing manifolds to investigate the following question of Gromov which asks: What is the weakest notion of convergence such that a sequence of Riemannian manifolds, M n j with scalar curvature R j ≥ κ subconverges to a limit M ∞ which may not be a manifold but has scalar curvature greater than κ in some suitably generalized sense? ...
The rigidity theorems of Marques-Neves and of Llarull, which show two different ways scalar curvature can characterize the sphere, have associated stability conjectures. Here we produce the first examples related to these stability conjectures. The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence. The second set of examples constructs sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume preserving intrinsic flat sense. In order to construct such sequences, we improve the Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount. Moreover, we are able to generalize both the sewing construction of Basilio, Dodziuk, and Sormani, and the construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit with no geodesics.
... We should also keep in mind the lack of rigidity discovered in work of Corvino [47], Lohkamp [96], Hang-Wang [68], and Brendle-Marques-Neves [32]. We should also keep in mind the Schoen-Yau and Gromov-Lawson tunnel constructions in [124] and [59], and the more recent sewing constructions of Basilio-Dodziuk-Sormani in [22] using those tunnels. In dimension 4 and up there is an intriguing set of examples by Lee-Naber-Neumayer in [87] using a new method altogether. ...
... as a consequence of (22). Conversely, if we start with curves C j : [0, 1] → M j with a uniform upper bound on their lengths, the Arzela-Ascoli Theorem implies that they have a subsequence which converges to rectifiable curves in C ∞ : ...
... The first Scalar Compactness Conjecture was suggested by Gromov in [65] and further refined by Sormani in [135] building upon work with Basilio and Dodziuk in [22] and work of Park-Tian-Wang in [115]. First recall the definition: ...
Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}. We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above. In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.
... In an appendix to an earlier paper [1] we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant positive sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature. ...
... Note that the earlier constructions in [3], [4], [9], and [8] did not give information about the size of the tunnels. In [1], we proved the theorem above for manifolds of three dimensions and constant positive sectional curvature. This was sufficient for constructing examples of sequences of manifolds of positive scalar curvature whose limits (under any reasonable notions of convergence) did not have positive generalized scalar curvature in the sense explained in [1]. ...
... In [1], we proved the theorem above for manifolds of three dimensions and constant positive sectional curvature. This was sufficient for constructing examples of sequences of manifolds of positive scalar curvature whose limits (under any reasonable notions of convergence) did not have positive generalized scalar curvature in the sense explained in [1]. Our result removes the restriction on the dimension and allows for variable sectional curvature. ...
Cambridge Core - Abstract Analysis - Analysis and Geometry on Graphs and Manifolds - edited by Matthias Keller
... The equality on the volume bound and the diameter bound in Conjecture 1.2 prevent collapsing and expanding. There are examples of Gromov-Hausdorff and intrinsic flat convergent sequences of Riemannian manifolds M j satisfying minA(M j ) → 0, in which the scalar curvature blows up to negative infinity [8]. ...
By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of 3-dimensional tori \(M_j\) that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form \(R_{g_{M_j}} \ge -1/j\). We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang–Lee, Huang–Lee–Sormani and Allen–Perales–Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus \((M, g_M)\) is replaced by a bound on the quantity \(-\int _T \min \{R_{g_M},0\} d{\mathrm {vol}_{g_T}}\), where \(M=\text {graph}(f)\), \(f: T \rightarrow \mathbb {R}\) and \((T,g_T)\) is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions \(n \ge 4\) as well.
... uses Hausdorff measure to replace volume does not behave well under convergence. In joint work of the authors with Dodziuk, we constructed a sequence of manifolds with positive scalar curvature which converged in the Gromov-Hausdorff and Intrinsic Flat sense to a limit space for which this limit is negative at a point [BDS18]. That example was constructed using a method we called "sewing along a curve" and the limit space was a standard three dimensional sphere in which one of the closed geodesics was "pulled to a point". ...
... It is somewhat technical if one does not already know the methods Gromov-Hausdorff and Intrinsic Flat convergence. A review of the necessary background can be found in [BDS18] so we do not repeat it here. ...
... In Section 2, we introduce our Method I for creating sequences of manifolds with positive scalar curvature that converge to pulled metric spaces. We begin by reviewing the construction of tunnels of positive scalar curvature found by Schoen-Yau [SY79a] and Gromov-Lawson [GL80] (cf the appendix to [BDS18]). We review also the method of sewing along a curve by placing the tunnels in a paired pattern along a fixed curve in a fixed manifold to create a new manifold with positive scalar curvature. ...
We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with "pulled regions". The examples created rigorously within using these methods were announced by us a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of "sewing along a curve" developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.
