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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 this paper we study non-negatively curved and rationally elliptic GKM4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds.

Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in [Wiemeler, Torus manifolds and non-negative curvature, arXiv:1401.0403] and was used there to obtain a classification of non-negatively curved torus 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.

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.

 Journal Mathematical Proceedings of the Cambridge Philosophical Society Link to preprint version

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.

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.

 Journal Int. Math. Res. Not. IMRN Volume 18 Pages 5786--5822 Link to preprint version Link to published version

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.

 Journal Bulletin of the London Math. Society Volume 50 Pages 96-107 Link to preprint version Link to published version

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.

 Journal Journal of Differential Geometry Link to preprint version

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).

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.

 Journal preprint arXiv Pages 11 pages Link to preprint version

We prove that the Teichmüller space of negatively curved metrics on a hyperbolic manifold M has nontrivial i-th rational homotopy groups for some i > dim M. Moreover, some elements of infinite order in the i-th homotopy group of BDiff(M) can be represented by bundles over a sphere with fiberwise negatively curved metrics.

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

The canonical map from the $$\mathbb{Z}/2$$-equivariant Lazard ring to the $$\mathbb{Z}/2$$-equivariant complex bordism ring is an isomorphism.

## Team Members

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

Dr. Mauricio Bustamante Londoño
Researcher
Binghamton University
bustamante(at)math.binghamton.edu

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

Dr. Georg Frenck
Researcher
Universität Augsburg
georg.frenck(at)uni-a.de

Dr. David González Álvaro
Researcher
Université de Fribourg
david.gonzalez.alvaro(at)upm.es

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

Jan-Bernhard Kordaß
Doctoral student
KIT Karlsruhe

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

Prof. Dr. Wilderich Tuschmann
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