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

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 Positive scalar curvature and Poincaré duality for proper actions

2019 ◽  
Vol 13 (4) ◽  
pp. 1381-1433
Author(s):  
Hao Guo ◽  
Varghese Mathai ◽  
Hang Wang
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Proper Actions ◽  
Positive Scalar

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.

Positive scalar curvature metrics via end-periodic manifolds

2020 ◽  
Vol 5 (3) ◽  
pp. 639-676
Author(s):  
Michael Hallam ◽  
Varghese Mathai
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

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