Project Description Convergence of Riemannian Manifolds

Home , Comparison theorem, Geometry Festival, Jeff Cheeger, Space form

1. Introduction

In 1967, Cheeger introduced the notion of converging sequences of Riemannian manifolds, proving that sequences of compact manifolds with uniformly bounded sectional curvature, 1 |sec (Mi) | ≤ K, and diameter, diam (Mi) ≤ D0, have subsequences which converge in the C k sense [15][16]. The C distance between a pair of diﬀeomorphic Riemannian manifolds, M1 and

M2, with metrics g1 and g2 respectively, is

n ∗ o k k (1) dC (M1,M2) := inf |ϕ (g2) − g1|C (g1) : diﬀeom ϕ : M1 → M2 .

k Here the infimum is taken over C diffeomorphisms ϕ : M1 → M2, so it is only defined for pairs of diffeomorphic manifolds. In 1981, Gromov introduced a weaker notion of convergence, an intrinsic Hausdorff convergence, proving that Riemannian manifolds with Ricci(M) ≥ (n − 1)H and diam(M) ≤ D have subsequences converging in this Gromov-Hausdorff sense to a compact geodesic metric space

[38]. Gromov’s distance is deﬁned by taking an inﬁmum over all isometric embeddings ϕi of the

Mi into any common metric space Z:

n Z o (2) dGH (M1,M2) = inf dH (ϕ1(M1), ϕ2(M2)) : isom ϕi : Mi → Z

Z where dH is the Hausdorﬀ distance between subsets in Z deﬁned by

Z n o (3) dH (A1,A2) := inf R : A1 ⊂ TR(A2),A2 ⊂ TR(A1) .

The Gromov-Hausdorff limits of Riemannian manifolds are no longer Riemannian manifolds but they are geodesic metric spaces, spaces with minimal geodesics running between all pairs of points. Gromov proved that sequences of Riemannian manifolds with uniform bounds on their diameter and on the number, N(r), of disjoint balls of radius r have subsequences which converge in the Gromov-Hausdorff sense to a compact limit space. By the Bishop-Gromov volume comparison theorem, sequences of manifolds with nonnegative Ricci curvature and diameter satisfy these conditions [38]. Greene-Petersen have shown that sequences of manifolds with a uniform upper bound on volume and a uniform geometric contractibility functions also satisfy Gromov’s conditions and thus have a converging subsequence [35]. In 1987, Fukaya introduced metric measure convergence [33]. Cheeger-Colding then proved that manifolds with nonnegative Ricci curvature have subsequences converging in the metric measure sense. Among other things, they proved the limit spaces are countable Hm rectifiable spaces. They also proved the limit spaces, viewed as metric measure spaces, have many properties in common with manifolds of nonnegative Ricci curvature [18]. Sturm and Lott-Villani extended 1 the notion of a Ricci curvature bound to general metric measure spaces using mass transport [78][51]. Recently Topping has been developing a notion of Ricci flow on this larger class of spaces in hopes of defining Ricci flow through a singularity [80]. Despite the immense success in applying these definitions of convergence to study manifolds with Ricci curvature bounds, there has been a need to introduce a weaker form of convergence to study sequences of manifolds which do not satisfy these strong conditions. Mathematicians studying manifolds with scalar curvature bounds and those interested only in sequences with an upper bound on volume and diameter without curvature bounds, need a weaker version of convergence. In particular, geometric analysis related to cosmology and the study of the spacelike universe requires a weaker form of convergence. Recently Stefan Wenger and I have introduced a new convergence that we call the intrinsic flat convergence [76][75]. It is based on the flat convergence of integral currents defined by Federer- Fleming to study minimal surfaces. We have rigorously defined the intrinsic flat distance between oriented Riemannian manifolds using Ambrosio-Kirchheim’s new notion of an integral current, T , on a metric space: which is effectively a definition of integration over a weighted oriented m m countably H rectifiable subset. Given an oriented Riemannian manifold, Mi , there is a natural notion of integration over Mi which defines an integral current Ti such that M(Ti) = vol(Mi) and ∂Ti is integration over the boundary of Mi. If ϕi : Mi → Z is an isometric embedding, then the pushforward ϕi#Ti is an integral current in Z of dimension m. Significant work on Ambrosio-Kirchheim’s notion of integral currents on metric spaces and the flat distance between them has been completed by my coauthor Stefan Wenger in [85][84]. Wenger and I have defined the intrinsic flat distance between Riemannian manifolds to be:

n Z o (4) dF (M1,M2) := inf dF (ϕ1#T1, ϕ2#T2): isom ϕi : Mi → Z where the ﬂat distance in Z is deﬁned as in Federer-Fleming

Z n o (5) dF (S1,S2) = inf M(A) + M(B): int curr A, B s.t. A + ∂B = S1 − S2 .

We have proven dF (M1,M2) = 0 iff there is an orientation preserving isometry between them. The limit spaces are called integral current spaces, and are oriented weighted countably Hm rectifiable metric spaces. An important integral current space is the 0 space, and collapsing sequences of manifolds, vol(Mj) → 0, converge to the 0 space. We also relate the intrinsic flat convergence of Riemannian manifolds to the weak convergence of integral currents [75]. I have presented this work at the Geometry Festival last Spring, as well as Columbia, Harvard, Dart- mouth, Urbana-Champagne, Johns Hopkins and soon Rutgers and U. Penn. The presentation is available on my website and has graphics depicting many examples we have proven converge in the intrinsic flat sense. 2 In an initial joint paper which has been accepted for publication, Wenger and I prove key results about the weak limits of integral currents needed to understand intrinsic flat convergence [76]. In some cases the intrinsic flat limits and Gromov-Hausdorff limits agree giving new insight into the rectifiability of the Gromov-Hausdorff limits. We prove the intrinsic flat limits and Gromov-Hausdorff limits agree for noncollapsing sequences of manifolds with nonnegative Ricci curvature and also for sequences of manifolds with uniform linear contractibility functions and uniform upper bounds on volume. As a consequence their Gromov-Hausdorff limits are countably Hm rectifiable metric spaces. This was already shown by Cheeger-Colding for the sequences with bounded Ricci curvature but is a new result for the sequences with the linear geometric contractibility hypothesis. With only uniform geometric contractibility functions that are not linear, Schul and Wenger have shown the limits need not be so rectifiable [76] and when there is no upper bound on volume, Ferry has shown the limit spaces need not even be finite dimensional [31]. In general the intrinsic flat and Gromov-Hausdorff limits do not agree: the intrinsic flat limit may be a strict subset of the Gromov-Hausdorff limit. If the Gromov Hausdorff limit is lower dimensional than the sequence, then the intrinsic flat limit is the 0 space (i.e. the manifolds disappear completely in the limit). Sequences may also disappear due to an effect we call cancellation. We have results which prevent this disappearance involving the filling volumes of spheres in the space [76][75]. This is discussed further within the proposal. The intrinsic flat convergence is weaker than Gromov-Hausdorff convergence. Sequences which do not converge in the Gromov-Hausdorff sense may converge in the intrinsic flat sense [75]. In fact, Stefan Wenger has a preprint proving that a sequence of oriented Riemannian manifolds with a uniform upper bound on diameter and on volume has a subsequence which converges in the intrinsic flat sense to an integral current space [86]. In this project, I propose to study all these various forms of convergence, their limit spaces and the properties of the Riemannian manifolds which are conserved under the convergence. I would like to continue and extend prior work studying geodesics in converging sequences and applications of convergence to the study of the spacelike universe. I would like to reexamine the notion of Ricci and scalar curvature on limit spaces and Ricci flow through singularities using the additional rectifiability provided by the intrinsic flat convergence. In the long term I would also be interested in investigating new and even weaker notions of convergence. After describing the results from my prior and current NSF support, I describe short and long term research plans for the next five years with specific conjectures as well as problems for doctoral students.

3 2. Prior Support

In the past ﬁve years I have been funded by two NSF grants. I begin with the ﬁrst:

NSF DMS-0102279, The Topology of Manifolds with Nonnegative Ricci Curvature, (June 1, 2001-May 31, 2006, $85,714.00): This grant covered travel, visitors and summer salary. The research began as a study of the topology of smooth manifolds with lower bounds on Ricci curvature and their Gromov-Hausdorff limits. It also explored applications of Gromov-Hausdorff convergence to cosmology. Finally I examined properties conserved under Gromov Hausdorff convergence related to the Laplace and length spectra. In the next four subsections I describe this research.

2.1. Topology of Smooth Riemannian Manifolds: At the time that I was awarded this grant, I had just proven a partial solution to Milnor’s conjecture that the fundamental group of a complete noncompact manifold with nonnegative Ricci curvature is finitely generated [53]. This conjecture had already been proven when the volume growth is maximal by Anderson and Li [3] [49]. I had proven the conjecture when the manifold has small linear diameter growth [68]. As a consequence, complete noncompact manifolds with linear volume growth or sublinear diameter growth have finitely generated fundamental groups. The exact constant needed to define small linear diameter growth was improved in work of Xu, Wang and Fang [89] and the results have been extended by Wylie in [88]. Zhongmin Shen and I then studied the higher dimensional homology of manifolds with nonnegative Ricci curvature without any assumption on diameter growth. Work of Perelman and Menguy demonstrated that the second homology of such spaces could be unbounded [56] [52].

However Yau had shown the Hn−1(M, R) is trivial when the Ricci curvature is positive [90]. Further work had been completed by Itokawa-Kobayashi using integral currents [46]. Applying results about noncontractible loops I had proven in [69] and techniques of algebraic topology,

Shen and I completely classiﬁed Hn−1(M, Z), proving it is trivial unless a double cover of the space splits isometrically. Recently, Zhongmin Shen and I wrote a survey article describing open problems concerning the topology of smooth manifolds with nonnegative Ricci curvature [65]. I presented this work in invited address at the 2006 Midwest Geometry Conference, the 2008 Southeast Geometry Conference and the 2008 UNAM Workshop on Global Geometry in Cuernavaca Mexico .

2.2. Topology of Gromov-Hausdorff Limits. During the tenure of this first grant, Riemann- ian geometers had become increasingly interested in the Gromov-Hausdorff limits of manifolds with nonnegative Ricci curvature. Using work of Perelman, Menguy had shown that the limit spaces of such sequences could have locally infinite topological type, even in the noncollapsed 4 setting where the limit space has the same Hausdorff dimension as the sequence [52]. Cheeger- Colding, on the other hand, wrote a series of papers proving, among other things, that the limit spaces are countably Hm rectifiable [18][19][20]. So it became natural to study the topology of these limit spaces. First, Guofang Wei and I proved that the Gromov-Hausdorff limits of manifolds with nonnegative Ricci curvature have universal covers [72]. Recall that a space with local infinite topological type need not have a universal cover. Keep in mind the Hawaii Ring which is created by joining at a point infinitely many circles of radii decreasing to zero [77]. Menguy’s example had infinitely many nonhomotopic spheres converging on a point [52]. In our second paper, Wei and I dealt with complete noncompact limits. We extended the theorems of Anderson and myself mentioned above regarding the fundamental groups of manifolds with nonnegative Ricci curvature to limit spaces [3][68]. We used the metric measure convergence techniques of Cheeger Colding [18] and my uniform cut lemma [68]. I presented this work in seminars at Stony-Brook, Johns Hopkins, Brown and Columbia and in a series of talks at the New Approaches to Curvature Summer School at Les Diablerets, Switzerland. Within this work, Wei and I developed a new concept called a δ cover, which is a covering space that does not unravel loops that fit in balls of radius δ [72]. The group of deck transforms of this cover is well controlled under Gromov-Hausdorff convergence. This notion and related notions have recently been further developed by topologists Plaut, Beretovsky-Plaut and Guillemette to explore questions completely unrelated to Ricci curvature or Riemannian geometry [57] [8] [59].

2.3. Application to Friedmann Cosmology: One of the most important applications for an abstract notion of distance is to prove the stability of a physical model. A physical model is stable if a small deviation in observable data leads only to a small deviation in the predictions given by the model. Different notions of distance may be used to measure these deviations. So I decided to investigate whether Gromov-Hausdorff distance could be used to study the stability of the space-like universe, which is a Riemannian manifold. In the Friedmann model of cosmology, it is observed that the spacelike universe is almost locally isotropic. By Schur’s Lemma, a Riemannian manifold which is locally isotropic has constant sectional curvature and is, by definition, a space form. Thus in the Friedmann model, the universe is assumed to be a space form (c.f. [32]). In fact, the spacelike universe is not locally isotropic: clouds of dust can cause weak gravitational lensing and heavy objects can cause strong gravitational lensing. Nevertheless, this model is used to predict the timing of the Big Bang and the expansion of the universe (c.f. [55]). One would like to be sure that the predictions of the model are still close to reality, despite the small deviation in the assumption of local isotropy. 5 Examining the problem mathematically, one first assumes only weak gravitational lensing, assuming the Riemannian manifold (the spacelike universe) to be smoothly close to having constant sectional curvature. Gribcov and Currier have shown that such a manifold need not be close to a space form in the smooth sense (the metric need not be close to such a metric) [36][26]. Schur’s Lemma is not stable. In 2004, I proved that Friedmann’s model of cosmology is stable when one uses the Gromov- Hausdorff distance between manifolds rather than the smooth distance [70]. That is, if a manifold is almost locally isotropic in a sense which allows for both weak and localized strong gravitational lensing, then this manifold is Gromov-Hausdorff close to a collection of space forms glued together at points. For example, a universe Gromov-Hausdorff close to two round spheres joined by a black hole is locally isotropic with strong localized gravitational lensing around the black hole. In order to prove this result I had to add an additional condition on the spacelike universe. This necessary condition allowed me to uniformly bound N(r) and apply Gromov’s Compactness Theorem. Without this condition, there is a counter example, a sequence of spaces which are not close to space forms that become increasingly locally isotropic (and in fact increasing hyperbolic). Note that an assumption of nonnegative Ricci curvature implies this condition [70][38]. In addition to Gromov’s work, I applied Grove-Petersen’s Arzela-Ascoli Theorem to control the divergence of geodesics in the limit space [39]. I developed a notion of a space called an exponential length space, defined conjugate points on such spaces, studied their universal covers and proved the universal cover was spherical, hyperbolic or Euclidean space using Birkhoff’s Theorem [9]. In recent work with Krishnan Shankar [64], we have extended the theory of conjugate points to arbitrary geodesic spaces relating this work to work of Alexander-Bishop on Alexandrov spaces [1][2]. I have presented this work at seminars in Princeton, Stanford, Dartmouth and Arizona, some of which were attended by cosmologists. I also presented it to physicists at the Seminar on Group Theoretic Methods in Theoretical Physics. When presenting the result at Stanford, Schoen and Huisken recommended that I use a weaker form of convergence than Gromov- Hausdorff convergence to extend the result to manifolds with nonnegative scalar curvature. Nonnegative scalar curvature is a natural condition on a spacelike universe, however, at that time, the weakest definition of convergence for a sequence of Riemannian manifolds was the Gromov-Hausdorff convergence. In this proposal, I plan to apply the intrinsic flat convergence and Wenger’s Compactness Theorem to extend these results.

2.4. The Covering Spectrum. When studying the convergence of Riemannian manifolds, it is important to understand what properties are conserved under the taking of a limit, and 6 what properties are continuous. Smoothly converging manifolds are always diffeomorphic to their limits and so the topology is conserved. The topology may increase or decrease under Gromov-Hausdorff and metric measure convergence. However, other properties are conserved. Cheeger-Colding have proven that in this setting the Laplace spectrum converges [20] and Ding has proven the heat kernal does as well [28]. Guofang Wei and I defined the notion of the covering spectrum of a metric space. This spectrum is a subset of the length spectrum defined using δ-covers [73]. In effect, the covering spectrum measures the sizes of one dimensional holes in a space, and it can be used to determine whether the metric space has a universal cover or not. We proved this spectrum is continuous with respect to Gromov-Hausdorff convergence. Intuitively holes may decrease in size to 0 and disappear, but they cannot suddenly appear or disappear. One key step in the proof was to prove a compactness theorem for the δ covers of compact collections of Riemannian manifolds. More recently we have proven similar results in the complete noncompact setting, effectively describing what happens to one dimensional holes under pointed Gromov-Hausdorff convergence [74]. I have presented this work in invited addresses at the Spectral Geometry Workshop at CRM in Montreal, the Bloomington Geometry Workshop, and the Texas Geometry Conference in 2007 as well as at a variety of differential geometry seminars. Naturally it is of interest to understand which Riemannian manifolds have the same covering spectra and what conditions determine such isospectrality. Recently DeSmit-Gornet-Sutton have posted a preprint on the arxiv extended Sunada’s method from Laplace spectral geometry [79] and work of Gordon [34], to determine when certain types of Riemannian manifolds are isospectral with respect to the covering spectrum [27].

2.5. The Length Spectrum. The length spectrum of a Riemannian manifold is the set of lengths of smoothly closed geodesics. On generic manifolds the Laplace spectrum determines the length spectrum. This was proven by Colin de Verdiere [24] and by Duistermaat-Guillemin [30]. Although the Laplace spectrum is continuous under smooth convergence, the length spectrum is not. There are examples of smooth manifolds converging to a smooth manifold with a suddenly appearing degenerate geodesic (one with a smoothly closed parallel vector field). Under Gromov- Hausdorff convergence, smoothly closed geodesics running between a pair of increasingly tiny holes can disappear in the limit. Zelditch was intrigued by the fact that Wei and I had shown that the covering spectrum (a subset of the length spectrum) converges under Gromov-Hausdorff convergence. He had been studying, among other things, the traces of billiard maps and their relation to eigenfunctions. Smoothly closed geodesics are periodic orbits in billiard maps. He suggested finding a larger 7 subset of the length spectrum which converges. This lead to my most recent paper funded by this NSF grant. In 2006, I defined the 1/k length spectra. The 1/k spectrum is the collection of lengths of closed geodesics which are minimizing on any subinterval whose length is 1/k of the total length of the closed geodesic. Such geodesics are naturally smoothly closed because we include intervals about the endpoint. Any smoothly closed geodesic is a 1/k geodesic for some value of k. In this paper I prove that the 1/k length spectra, for fixed k, is continuous with respect to Gromov-Hausdorff convergence of the manifolds. Lengths may converge to 0 and disappear but they cannot suddenly disappear or appear as occurs for the length spectrum [71]. The systole of a manifold is the length of the shortest noncontractible closed geodesic, and thus an element of the 1/2 length spectrum. Relationship to work by Pu [58] , Croke-Katz [25], Bavard and Sakai [7] [62], Rotman [60] , Sabourau [61] , and Nabutovsky-Rotman [54] on systoles is explored in the paper. The paper includes a number of open questions easily accessible to doctoral students who need to complete a first research paper. Ho, a doctoral student at Penn State, has already published a solution to one of the problems [44].

NSF MSP 0832247 Mathematics Teacher Transformation Institutes (October 1, 2008 - September 30, 2011, $3,093,602.00).

2.6. Geometry for Teachers. Currently I am one of ﬁve coPIs on this Mathematics Science Partnership program. Here I have worked with three other research mathematicians, Melvyn Nathanson, Nikola Lakic and Robert Schneiderman, to design a sequence of mathematics courses to enhance the background of certiﬁed public high school teachers in New York City. My focus has been on the new New York State curriculum which now includes a year of Euclidean Geometry taught both with a modern selection of Euclidean axioms as well as transformations, symmetries and coordinate geometry. At this stage of the grant, the courses are already designed and I am teaching the MTTI geometry classes as part of my ordinary teaching at Lehman College. I am not involved in the education research related to this grant beyond the testing of the teacher participants’ ability to write and explain proofs. The incoming assessments have been completed. The research is being conducted by my colleagues in the Education Department at Lehman College, Dr. Serigne Gningue (coPI) and Dr. Richard Peach. The grant covers the cost of a project director who handles all the logistics and the cost of education doctoral students and postdocs who assist with the education research. It pays for released time to develop the new courses. It does not support mathematical research or doctoral students in mathematics.

8 3. Research Proposal

3.1. Ricci and Scalar Curvature on Limit Spaces: After proving the metric measure convergence of Riemannian manifolds with nonnegative Ricci curvature, Cheeger-Colding proved a number of properties on the limit spaces of these manifolds, including the volume comparison theorem and the splitting theorem. Subsequently, Sturm and Lott-Villani defined a notion of nonnegative Ricci curvature on metric measure spaces in general using optimal transport. Many properties have been proven to hold on these spaces including the volume comparison theorem. The splitting theorem, however, does not hold in their generalized setting. In fact the taxi cab metric on the plane satisfies their notion of nonnegative Ricci curvature but does not satisfy the splitting theorem. As a consequence of the first paper with Stefan Wenger, we now understand that there is more structure on the limit spaces and have a new perspective as to how to view this structure [76]. In fact the limit spaces are described in our preprint [75] as metric spaces (X, d) with an integral current structure T . The current structure T is an integral current in X of the same dimension as X and defines a notion of orientation and weight on X. The mass measure ||T || agrees with the measure found under metric measure convergence and also agrees with the Hausdorff measure of the space when we have a noncollapsing sequence of manifolds with nonnegative Ricci curvature. The space X is countably Hm rectifiable and thus has a set of Lipschitz charts as well as a notion of a metric differential. The notion of the metric differential was developed in work of Korevaar-Schoen [48] and Kirchheim [47]. We call spaces with all these properties including the integral current structure, integral current spaces. I believe that the integral current structure on the limit spaces can be used to define a notion of a Ricci differential, similar to the metric differential. This Ricci differential could play the role of a Ricci tensor just as the metric differential has played the role of the metric tensor:

Conjecture 1. An integral current space (X, d, T ) has a natural notion of a Ricci tensor deﬁned almost everywhere using the metric diﬀerential and integral current structure.

Conjecture 2. An integral current space which has nonnegative Ricci curvature with respect to this definition has nonnegative Ricci curvature of the correct dimension defined using optimal transport and the mass measure ||T || and thus satisfies volume comparison and other properties they have proven.

Conjecture 3. An integral current space with nonnegative Ricci curvature deﬁned using this tensor satisﬁes the splitting theorem: it splits isometrically if it contains a line.

Note that for Conjecture 3 to be true, our definition of the Ricci differential on the taxi cab plane cannot have nonnegative Ricci curvature as that space does not split isometrically. So this 9 would be moving towards a more restricted notion of nonnegative Ricci curvature that excludes the taxicab plane. In [75], Wenger and I have already studied the taxicab space in some detail. One interesting contrast between the taxicab space and the integral current spaces which are achieved as limits in the work of Cheeger-Colding, is that the mass measure and the Hausdorff measure do not agree on the taxicab space while they do agree on the Cheeger-Colding’s limit spaces. Our preprint explores the relationship between Hausdorff measure, mass measure and the metric differential. William Wylie and Guofang Wei have written a paper exploring many equivalent notions of Ricci curvature on Riemannian manifolds. There is of course the usual notion as the trace of the sectional curvatures, but one can also define it using the Bochner formula. By directly relating the Ricci curvature to the Laplacians of functions, one can more directly see how on a Riemannian manifold the Ricci tensor controls both volume and splitting. I propose to work with William Wylie to explore the possibility of defining a Ricci tensor on integral current spaces using this approach combined with the notion of the metric differential and work of Cheeger on gradiants. The Bochner formula involves Laplacians (which can be defined using the mass measure), gradiants (which can be defined using the metric differential and work of Cheeger) and also the Hessian. We expect the Hessian term to be the most challenging aspect. To define a notion of nonnegative Ricci we do not really need to control the Hessian but it is an important part of the splitting theorem. So we will need some sort of weak notion of Hessian to complete the full conjecture but not to complete the first two parts. Scalar curvature, on the other hand, is easily defined on metric measure spaces and on integral current spaces just by examining the limits of volumes of balls whose radii decrease to zero. A natural definition for a Ricci tensor would have to agree with this definition, as scalar curvature is the trace of the Ricci tensor. On a preliminary level one would just like to confirm any new notion of nonnegative Ricci curvature implies nonnegative scalar curvature. On a second level, it would be nice if one could determine the scalar curvature as the trace of a Ricci tensor. Finally one would like Ricci curvature bounds, scalar curvature bounds or at least total scalar curvature bounds to be conserved or at least lower semicontinuous. This needs to be investigated further both for sequences of Riemannian manifolds converging in the metric measure sense and for sequences converging in the intrinsic flat sense.

3.2. Ricci Flow: One source of the demand for a notion of a Ricci tensor on limit spaces, is to extend Hamilton’s Ricci flow through singularities. Mean curvature flow, which is similar to Ricci flow in many senses, does have a notion of a flow through a singularity. This weak notion of mean curvature flow is called the Brakke flow which is defined using varifolds. Recently White related the notion of varifold convergence and integral current convergence proving that 10 the Brakke flow is continuous with respect to flat convergence when the varifolds are viewed as integral currents [87]. Thus it becomes natural to study a Ricci flow defined using integral current spaces which is continuous with respect to intrinsic flat convergence:

Conjecture 4. One can define Ricci flow through singularities by extending the notion of Ricci flow to integral current spaces.

Clearly if the Ricci differential is developed as described in the prior section it might be used to define a flow of the metric differential which naturally extends the Ricci flow. However, we need not complete the proofs of Conjectures 1-3 to proceed with this project. In fact, ideas related to Ricci flow might help us to define an appropriate notion of Ricci curvature by closely examining whether the work of White, Brakke and Ilmanen define a weak notion of mean curvature on integral currents in Euclidean space. In other words, it might be easier to extend the notion of Ricci flow than to define the Ricci tensor. The key missing ingredient that has kept people from directly extending the work on Brakke flow to better understand Ricci flow is that there has never been a notion of an intrinsic varifold space and intrinsic varifold convergence. There is no metric measuring the distance between varifolds, just the natural notion of weak convergence against test functions. In contrast, Federer- Fleming showed that integral currents not only have a weak convergence but a flat distance between them. It was this distance that allowed Stefan Wenger and I to extend the notion of flat convergence to an intrinsic flat convergence. With the recent work of White, one should expect yet another natural connection between the Ricci and mean curvature flows. Obviously preliminary investigations need to be made, exploring the convergence of key examples, and this can be done by doctoral students. There are already precisely studied examples approaching singularities for the Ricci flow. In addition to Hamilton’s many examples, there are the precisely described neck pinch singulaties in work of Angenent-Knopf [5] [4]. Other key examples appear in the books of Chow-Knopf and Chow-Lu-Ni [22] [23]. Degenerate singularities are studied in detail by Ding in [29]. There is also work of Simon describing how one should Ricci flow from a C0 metric [67]. So the doctoral students could verify that these examples are in fact continuous with respect intrinsic flat convergence, which would be an easy project, and then investigate how the integral current structure flows in these settings which would require deeper understanding. Note that once one has a sequence of manifolds converging smoothly on a compact domain avoiding the boundary, one needs only show that the region without smooth convergence has volume decreasing to 0 to obtain an intrinsic flat limit. So this aspect of the investigation is quite simple. The more difficult aspect is understanding the properties of the integral current structure on the singular limit space. 11 Topping-McCann and Topping have proposed an approach to Ricci flow through singularities using the mass transport approach to Ricci curvature [80][81]. Roughly the idea is to define a Ricci flow on metric measure spaces as a kind of infimum over all possible flows solving the equation with an inequality. Remember mass transport only defines a lower bound for Ricci curvature and not a complete Ricci tensor. This appears to define a unique flow, which may perhaps be a surprising fact or it may be selecting one of many flows. My doctoral student, Michael Munn, who graduated in May 2008, is working with Topping at Warwick University on related problems. It would be important to investigate whether the flows defined by Topping [80] are continuous with respect to intrinsic flat convergence. Naturally one would need to show that an integral current space flows into an integral current space, and this would probably be easily done since Topping has already proven continuity in the metric measure sense, which implies Gromov-Hausdorff convergence. So by the preprint with Wenger, we already know a subsequence converges in the intrinsic flat sense. We have theorems describing exactly when these limits agree. One would only have to investigate whether the integral current structure, T , and not just the mass measure, ||T ||, allows one to more easily understand the solutions Topping has constructed. Naturally one aspect of both this project is to see if there is a local test one can apply to the current structure T to quickly verify whether its mass ||T || satisfies the optimal transport notions of nonnegative Ricci curvature. Is there a simpler solution of the optimal transport problem when the space is an integral current space and can one more quickly verify that the solutions have convex entropy? Is there a condition on T which is essentially equivalent to the notion of Ricci ≥ H on a manifold that allows one to quickly confirm the space satisfies the optimal transport notion of Ricci ≥ H in the dimension of the space? That leads us directly back to the first part of the proposal and Conjectures 1-3.

3.3. Scalar Curvature: I was originally interested in developing a weaker notion of convergence of Riemannian manifolds after a discussion with Tom Ilmanen in 2006. He proposed the following:

Conjecture 5. There should be a weak notion of convergence such that sequences of three dimensional Riemannian manifolds with positive scalar and no interior minimal surfaces have a natural compactness theorem.

Ilmanen presented a number of examples which he felt should converge and described what their limit spaces should be. In these explicit examples the limit spaces were three dimensional Riemannian manifolds but the sequences did not converge in the smooth or the Gromov- Hausdorff sense. At first I had discussions with Dimitri Burago and then also Sergei Ivanov 12 about possibly developing a notion of area convergence which would match Ilmanen’s examples. In 2006 Wenger and I discovered we could define the intrinsic flat convergence and Wenger felt he could prove a compactness theorem. So we developed the notion and Wenger proved his compactness theorem which only involves diameter, volume and the volume of the boundary. It would seem then that we have already proven Ilmanen’s conjecture. However, clearly it is not enough just to define a weak convergence. We need some properties for our convergence which will prove useful to the study of manifolds with positive scalar curvature. As mentioned earlier it is imperative to show the scalar curvature bound is conserved in the limit. It is also important to understand properties related to the positive mass theorem and the Penrose inequality [63][10][45]. Do these theorems hold on integral current spaces with an appropriate notion of nonnegative scalar curvature? Can we define the ADM mass of an integral current space using its current structure? These are long term questions. The conservation of nonnegative scalar curvature in the limit may best be proven by using multiple equivalent definitions of scalar curvature using the integral current structure. The notion of quasilocal mass may be of some help in this investigation. While Bartnik’s definition in [6] may be difficult to handle, the notion developed by Shi-Tam, Liu-Yau and Wang-Yau in [66][50] [83] and [82] may be more tractable. It is crucial to have some kind of comparison theorem for scalar curvature and their work leads in this direction.

3.4. Limits of Geodesics: As mentioned above, Gromov proved that minimizing geodesics converge to minimizing geodesics when a sequence of manifolds converges in the Gromov- Hausdorff sense, but there are more subtle issues involved when studying closed geodesics. Many related questions appear in the final paper described in the prior support. Wenger and I have observed that when a sequence of Riemannian manifolds converges in the intrinsic flat sense, the limit space need not be a geodesic space. It may not even be a connected metric space. This can occur even when restricting to sequences three dimensional Riemannian manifolds with a uniform positive lower bound on scalar curvature [75]. Nevertheless, after a discussion with Ilmanen, we believe the following conjecture is true:

3 3 Conjecture 6. If a sequence of oriented Mj with Scal(Mj ) ≥ K > 0 and no interior minimal surfaces converges in the intrinsic ﬂat sense to an integral current space, M∞, then M∞ is a geodesic metric space.

The condition requiring no interior minimal surfaces is natural for spacelike manifolds in general relativity and is often added in theorems concerning three manifolds with positive scalar curvature. This includes work related to the positive mass conjecture and the Poincare inequality.

Recall that Wenger and I have proven that when we assume Ricci(Mj) ≥ 0 then the intrinsic

flat and Gromov-Hausdorff limits agree and so M∞ is a geodesic metric space. To prove a limit 13 space is a geodesic space directly, we need only show the midpoints of geodesics do not disappear unless an endpoint disappears. In the examples we constructed with positive scalar curvature and disappearing midpoints, there were always interior minimal surfaces near those midpoints. In [76] and [75] we prove points do not disappear in the limits when there are sufficiently strong uniform bounds on the filling volumes of spheres around the points. Here we refer to Gromov’s filling volume [37]. These bounds can be achieved either using local contractibility properties as in Gromov’s paper or using Lipschitz extensions of distance functions. So far we applied them to control the limits of manifolds with nonnegative Ricci curvature and the limits of manifolds with uniform linear geometric contractibility functions, and we believe similar techniques can be used for three manifolds with no interior minimal surfaces. While the three dimensional positive scalar curvature setting is of the most interest to those applying the results to cosmology, it is also of interest to find other conditions on a sequence of Riemannian manifolds which guarantee the convergence of minimizing geodesics.

3.5. Applications to Cosmology: As mentioned in the section on prior support, I have proven that a Riemannian manifold which is almost isotropic in a sense which allows for weak gravitational lensing and localized strong gravitational lensing is Gromov-Hausdorﬀ close to a metric space built from space forms joined at points [70]. In that paper, I had an additional condition on the space related to Ricci curvature. Schoen and Huisken had recommended that I extend the result to a setting where an assumption on scalar curvature is assumed in the place of Ricci curvature but the extension required a weaker notion of convergence. So I propose:

Conjecture 7. A sequence of Riemannian manifolds with nonnegative scalar curvature that is increasingly almost isotropic in a sense which allows for weak and localized strong gravitational lensing converges in the intrinsic ﬂat sense to a collection of space forms joined at points.

In fact examples revealing that deep but thin gravitational wells disappear under intrinsic flat convergence indicate that I might weaken the hypothesis requiring the localization of the strong gravitational lensing allowing long thin wormholes. Then one might expect the intrinsic flat limit to consist of disconnected set of space forms not necessarily joined at points. The proof of my original theorem in [70] strongly depended upon the convergence of geodesics, however, it may be possible to instead view the local isotropies as groups of almost isometries of balls. In [75], Wenger and I have shown that balls in Riemannian manifolds converge in the intrinsic flat sense to balls in the limit spaces. One might then use the almost isometries to control the limits of the balls and then show that the limit balls have a large class of isometries. Rather than completing the proof using conjugate points of geodesics, one might instead extend known results about metric spaces which are locally isotropic. 14 3.6. Other Projects: In addition to the projects described in detail above, I expect to continue projects related to Gromov-Hausdorff and metric measure convergence. Guofang Wei and I have a notion we call the rescaled covering spectrum which we believe will interact well with tangent cones at infinity. Krishnan Shankar and I have planned work relating to our first paper, closed geodesics and conjugate points on Gromov-Hausdorff limits. These projects follow naturally from work described under Prior Support. In the long term I am also interested in investigating other possible definitions of convergence. I may return to the work with Burago and Ivanov concerning the very weak notion of area convergence. They already published preliminary promising work in that direction in [11]. It is possible this notion will have applications in settings where intrinsic flat convergence does not work well. Continuity of area is very important in a number of settings and it does not hold under metric measure or intrinsic flat convergence without additional conditions.

4. Broader Impacts

The primary impact of the proposed research is to Cosmology and the study of General Relativity. It will also provide new insight into a number of problems in Geometric Analysis, a field which has had major influence on mathematics in this century. I write my papers on a level which can be understood by doctoral students including relationships with the work of other mathematicians so that the work can be applied by the nonexpert. I’ve designed webpages describing important modern mathematics on the level of a nonmathematician, edited a popular mathematics book, Poincare’s Prize written by Szpiro and worked as the AMS Media Contact for Riemannian Geometry. At the CUNY Graduate Center, I have three doctoral students that I hope to fund with this proposal. Sajjad Lakzian expects to complete his oral examination on Ricci flow this spring. He is currently taking a course at Columbia with Hamiton and a course at Courant with Kliener on this topic. Jorge Basilio just transfered to CUNY this Fall to work with me on Riemannian Geometry. Erica Fells is a Bridge to the Doctorate student who completed a masters course in differential geometry with me when she was an undergraduate at Lehman College. She has completed undergraduate research applying metrics on graphs to paleontology with Katherine StJohn funded by the Louis Stokes Alliance for Minority Participation. I have requested funding for only two doctoral students as Sajjad will be graduating when Erica begins doctoral research. CUNY is a minority serving institution with students of all backgrounds and nationalities. In addition to working with doctoral students, I regularly teach analysis to advanced undergraduate math majors and geometry to future math teachers. I also teach experienced math teachers in the Mathematics Teacher Transformation Institute and often teach college now precalculus courses for local public high school students through the Lehman College Math Circle.

15 References

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