Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

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Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics

Oberwolfach Reports ◽

10.4171/owr/2017/3 ◽

2018 ◽

Vol 14 (1) ◽

pp. 133-166

Author(s):

F. Thomas Farrell ◽

Wilderich Tuschmann

Keyword(s):

Moduli Spaces ◽

Riemannian Metrics

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An index theorem for end-periodic operators

Compositio Mathematica ◽

10.1112/s0010437x15007502 ◽

2015 ◽

Vol 152 (2) ◽

pp. 399-444 ◽

Cited By ~ 1

Author(s):

Tomasz Mrowka ◽

Daniel Ruberman ◽

Nikolai Saveliev

Keyword(s):

Scalar Curvature ◽

Moduli Spaces ◽

Differential Operators ◽

Riemannian Metrics ◽

Index Theorem ◽

Positive Scalar Curvature ◽

Fredholm Theory ◽

Eta Invariant ◽

First Order ◽

Periodic Operators

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

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On the topology of moduli spaces of non-negatively curved Riemannian metrics

Mathematische Annalen ◽

10.1007/s00208-021-02327-y ◽

2021 ◽

Author(s):

Wilderich Tuschmann ◽

Michael Wiemeler

Keyword(s):

Sectional Curvature ◽

Moduli Spaces ◽

Riemannian Metrics ◽

Rational Homotopy ◽

Analogous Statement ◽

Open Manifolds ◽

Negatively Curved ◽

Space Of Riemannian Metrics ◽

Negative Sectional Curvature ◽

Infinite Sequences

AbstractWe study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli 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 infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

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