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Spaces and Moduli Spaces of Riemannian metrics with curvature bounds on compact and non-compact manifolds (II)

This project consists of three subprojects based in Augsburg, Karlsruhe, and Fribourg (Switzerland), respectively.

1. New directions in positive scalar curvature geometry. We will study obstructions to and constructions of positive scalar curvature metrics on manifolds of low dimensions, manifolds with finite fundamental groups, manifolds with Baas-Sullivan singularities and non-compact manifolds. We will work on several versions of index theoretic obstructions on non-compact manifolds, relying on the coarse geometry approach of Roe and on the index theory of Dirac operators of Callias type. We will continue our research aiming at the construction of non-zero classes in higher homotopy groups of spaces and moduli spaces of positive scalar curvature metrics on non-compact manifolds.
2. Moduli spaces of Riemannian metrics: topology, topologies, and compactifications. We will investigate moduli spaces of Riemannian metrics of non-negative Ricci curvature on closed and open manifolds, aiming in particular at constructing first examples of simply connected manifolds for which these spaces have higher non-trivial rational cohomology and homotopy groups. Given both a manifold $$M$$, or a certain class of such $$M$$, and a suitable set of curvature conditions $$\mathcal C$$, we will study the set of all isometry classes of such metrics satisfying  $$\mathcal C$$, equipped with the (pointed) Gromov-Hausdorff or $$C^{k,\alpha}$$-topologies, and its closures and compactifications.
3. Moduli spaces of metrics with lower curvature bounds. We will continue our study of moduli spaces via $$\eta$$-invariants, with focus on metrics of nonnegative sectional or positive Ricci curvature and on dimensions not covered by previous results. We will try to define Kreck-Stolz-type invariants for new classes of closed manifolds including simply connected manifolds of dimension $$\neq 4k-1$$. We also want to look at equivariant refinements of these invariants and investigate how these can be used to analyze the moduli space of invariant metrics with lower curvature bounds using techniques such as fixed point formulas in index theory, rigidity and equivariant bordism theory.

## Publications

Based on  Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of  spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over  the sphere bundle $\mathbb{S}( \mathbb{R} \oplus V)$  in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.

 Journal São Paulo Journal of Mathematical Sciences (to appear) Link to preprint version

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

 Link to preprint version
• These are the refereed proceedings of the 2019 'Australian-German Workshop on Differential Geometry in the Large' which represented a cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks treating geometric evolution equations, structures on manifolds, non-negative curvature and topics in Kähler, Alexandrov and Sasaki geometry as well as differential topology.

 Journal London Mathematical Society Lecture Notes Series Publisher Cambridge University Press Book Differential Geometry in the Large Volume 463 Pages 398 Link to preprint version Link to published version
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## Team Members

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

Prof. Dr. Bernhard Hanke