Project Description Convergence of Riemannian Manifolds 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 jsec (Mi) j ≤ 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 j' (g2) − g1jC (g1) : diffeom ' : M1 ! M2 : k Here the infimum is taken over C diffeomorphisms ' : M1 ! M2, so it is only defined for pairs of diffeomorphic manifolds. In 1981, Gromov introduced a weaker notion of convergence, an intrinsic Hausdorff conver- gence, 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 'i of the Mi into any common metric space Z: n Z o (2) dGH (M1;M2) = inf dH ('1(M1);'2(M2)) : isom 'i : 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 con- verge 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 'i : Mi ! Z is an isometric embedding, then the pushforward 'i#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 ('1#T1;'2#T2): isom 'i : 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].