15

Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

The project concentrates on three major issues.

  1. The space of positive scalar curvature metrics. Let \(M\) be a non-compact connected spin manifold admitting a complete metric of uniformly positive scalar curvature. The main goal is to construct non-zero classes in higher homotopy groups of \({\mathcal R}^{scal\ge\epsilon >0}(M)\), the space of complete metrics of uniformly positive scalar curvature, and related moduli spaces.
  2. Fiber bundles with geometric structures and spaces of Riemannian metrics. Given a smooth bundle \(M\to E\to B\), one wants to investigate when there exists a Riemannian metric on the vertical tangent bundles (viewed as a smoothly varying family of metrics on the fibres) whose restriction to each fibre satisfies some specific curvature bounds like, e.g., being almost flat or (almost) nonnegatively (Ricci) curved. Furthermore, the goal is to study and compare different topologies on (moduli) spaces of Riemannian metrics and extend useful known results.
  3. Moduli spaces for nonnegative sectional and positive Ricci curvature. The aim is to study moduli spaces of metrics of nonnegative sectional curvature and / or positive Ricci curvature and to construct new examples of manifolds with disconnected moduli spaces. The plan is to find new invariants of moduli spaces and to give applications to non-compact manifolds of nonnegative sectional curvature, in particular, to define Kreck-Stolz invariants for new classes of closed manifolds and compute \(\eta\)-invariants using various techniques, e.g., Lefschetz fixed point formula in APS-index theory, rigidity and bordism theory.

     


    Publications

    In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group G for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.

     

    Related project(s):
    11Topological and equivariant rigidity in the presence of lower curvature bounds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive.

    In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: "MOD-formal actions" and "actions of formal core".

    We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch--Brown models, A∞-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature.

    A major motivation for the new definitions was that an almost free action of a torus Tn↷X possessing any of the two new properties satisfies the toral rank conjecture, i.e. dimH∗(X;Q)≥2n. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization.

    Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7.

     

    Journalto appear in Transactions of the AMS
    Link to preprint version

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    In this article, we are interested in the question whether any complete contractible 3-manifold of positive scalar curvature is homeomorphic to \(\mathbb{R}^3\). We study the fundamental group at infinity, \(\pi^\infty_1\), and its relationship with the existence of complete metrics of positive scalar curvature. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial \(\pi_1^\infty\) is homeomorphic to \(\mathbb{R}^3\).

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    In this work we prove that the Whitehead manifold has no complete metric of positive scalar curvature. This result can be generalized to the genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive
    scalar curvature metrics on closed simply connected manifolds in dimensions at least 
    five appears on spin manifolds, and is given by the non-vanishing of the \(\alpha\)-genus 
    of Hitchin. 
    When unobstructed we will in this paper realise  a positive scalar curvature metric by an 
    immersion into Euclidean space whose dimension is uniformly close to the classical Whitney 
    upper-bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure 
    for constructing positive scalar curvature metrics.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.

     

    Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimensions at least five which have odd order abelian fundamental groups, are non-spin and  atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

     

     

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    Let N be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold M. In this paper, we study the extent to which N admits as much symmetry as M. Our main results are examples of N that exhibit two extremes of behavior. On the one hand, we find N with maximal symmetry, i.e. Isom(M) acts on N by isometries with respect to some negatively curved metric on N. For these examples, Isom(M) can be made arbitrarily large. On the other hand, we find N with little symmetry, i.e. no subgroup of Isom(M) of "small" index acts by diffeomorphisms of N. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    For a closed, connected direct product Riemannian manifold (M,g)=(M1×⋯×Ml,g1⊕⋯⊕gl), we define its multiconformal class [[g]] as the totality {f12g1⊕⋯⊕fl2gl} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a function fi2>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (Mi,gi) does, under the technical assumption dimMi≥2. We also show that, even in the case where every factor (Mi,gi) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We extend two known existence results to simply connected manifolds with

    positive sectional curvature: we show that there exist pairs of simply

    connected positively-curved manifolds that are tangentially homotopy equivalent

    but not homeomorphic, and we deduce that an open manifold may admit a pair of

    non-homeomorphic simply connected and positively-curved souls. Examples of such

    pairs are given by explicit pairs of Eschenburg spaces. To deduce the second

    statement from the first, we extend our earlier work on the stable converse

    soul question and show that it has a positive answer for a class of spaces that

    includes all Eschenburg spaces.

     

    JournalMathematical Proceedings of the Cambridge Philosophical Society
    Link to preprint version

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.

    The main application is a general approximation result by sections which have very restrictive local properties an open dense subsets. This shows, for instance, that given any K∈R every manifold of dimension at least two carries a complete C^1,1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course this is impossible for C^2-metrics in general.

     

    Related project(s):
    5Index theory on Lorentzian manifolds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

     

    JournalInt. Math. Res. Not. IMRN
    Volume18
    Pages5786--5822
    Link to preprint version
    Link to published version

    Related project(s):
    11Topological and equivariant rigidity in the presence of lower curvature bounds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We show that in each dimension 4n+3, n>1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

     

    JournalBulletin of the London Math. Society
    Volume50
    Pages96-107
    Link to preprint version
    Link to published version

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\mathbb{R}^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is "stable under tangential homotopy equivalence": if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to~\(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.

     

    JournalJournal of Differential Geometry
    Link to preprint version

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds and construct, in particular, the first classes of manifolds for which these spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. An analogous statement holds for spaces of non-negative Ricci curvature metrics in every dimension at least eleven (respectively, twelve).

     

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

    Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M.

     

    Journalpreprint arXiv
    Pages11 pages
    Link to preprint version

    Related project(s):
    15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds


    Team Members

    Prof. Dr. Christoph Böhm
    Researcher
    Universität Münster
    cboehm(at)uni-muenster.de

    Dr. Mauricio Bustamante Londoño
    Researcher
    Universität Augsburg
    mauricio.bustamantelondono(at)math.uni-augsburg.de

    Prof. Dr. Anand Dessai
    Project leader
    Université de Fribourg
    anand.dessai(at)unifr.ch

    Dr. David González Álvaro
    Researcher
    Université de Fribourg
    david.gonzalezalvaro(at)unifr.ch

    Prof. Dr. Bernhard Hanke
    Project leader
    Universität Augsburg
    hanke(at)math.uni-augsburg.de

    Jan-Bernhard Kordaß
    Doctoral student
    KIT Karlsruhe
    kordass(at)kit.edu

    Prof. Dr. Uwe Semmelmann
    Researcher, Project leader
    Universität Stuttgart
    uwe.semmelmann(at)mathematik.uni-stuttgart.de

    Prof. Dr. Wilderich Tuschmann
    Project leader
    Karlsruher Institut für Technologie
    wilderich.tuschmann(at)kit.edu

    Dr. Jian Wang
    Researcher
    Universität Augsburg
    jian.wang(at)math.uni-augsburg.de

    Dr. Masoumeh Zarei
    Researcher, Project leader
    Universität Augsburg
    Masoumeh.Zarei(at)math.uni-augsburg.de

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