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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 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.

 

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

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.

 

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

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

 

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

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Team Members