scholarly journals The action of the mapping class group on metrics of positive scalar curvature

2021 ◽  
Author(s):  
Georg Frenck
Keyword(s):  
Scalar Curvature ◽  
Mapping Class Group ◽  
Class Group ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Mapping Class ◽  
High Dimensional ◽  
Positive Scalar ◽  
Simply Connected

AbstractWe present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds

2018 ◽  
Vol 12 (04) ◽  
pp. 897-939 ◽  
Author(s):  
Simone Cecchini
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Complete Riemannian Manifold ◽  
Positive Scalar Curvature ◽  
Type Operator ◽  
Spin Manifold ◽  
Compact Set ◽  
Positive Scalar ◽  
C Algebras

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert [Formula: see text]-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when [Formula: see text] is a closed spin manifold, we show that if the cylinder [Formula: see text] carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on [Formula: see text] must vanish.

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

scholarly journals Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

2021 ◽  
Vol 6 (2) ◽  
pp. 319-356
Author(s):  
Hao Guo ◽  
Peter Hochs ◽  
Varghese Mathai
Keyword(s):  
Scalar Curvature ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Proper Actions ◽  
Positive Scalar

scholarly journals Einstein Four-Manifolds With Self-Dual Weyl Curvature of Nonnegative Determinant

2019 ◽  
Author(s):  
Peng Wu
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
The Self ◽  
Weyl Curvature ◽  
Positive Scalar ◽  
Four Manifolds ◽  
Simply Connected

Abstract We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

scholarly journals An effective algebraic detection of the Nielsen–Thurston classification of mapping classes

2014 ◽  
Vol 07 (01) ◽  
pp. 1-21 ◽  
Author(s):  
Thomas Koberda ◽  
Johanna Mangahas
Keyword(s):  
Mapping Class Group ◽  
Word Length ◽  
Class Group ◽  
Mapping Class ◽  
Finite Cover ◽  
Generating Set ◽  
Reduction System ◽  
Conjugacy Separability ◽  
Mapping Classes

In this paper, we propose two algorithms for determining the Nielsen–Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod (S), then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π1(S) which allows us to lift the action of ψ to a finite cover [Formula: see text] of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on [Formula: see text] to determine the Nielsen–Thurston classification of ψ.

scholarly journals On the classification of mapping class actions on Thurston's asymmetric metric

2013 ◽  
Vol 155 (3) ◽  
pp. 499-515 ◽  
Author(s):  
L. LIU ◽  
A. PAPADOPOULOS ◽  
W. SU ◽  
G. THÉRET
Keyword(s):  
Finite Type ◽  
Mapping Class Group ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Actions ◽  
The One

AbstractWe study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil–Petersson metric.

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