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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

We show that local deformations of solutions to open partial differential relations near suitable subsets can be extended to global solutions, 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 ordinary differential inequalities, convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application concerns Riemannian metrics. We prove an approximation result which implies, for instance, that every compact surface carries a C^{1,1}-metric with Gauss curvature ≥1 a.e. on a dense open subset. For C^2-metrics this is, of course, impossible if the genus of the surface is positive.

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

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 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.

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.