Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

Guo, Hao

doi:10.1007/s12220-019-00249-5

Cited by 11 publications

(46 citation statements)

References 33 publications

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“…is a well-defined vector in L 2 (S) ⊗ H. By computations similar to those in the proof of Proposition 5.4 in [15], one sees that v L 2 (S)⊗H equals…”

Section: Properties Of the G-callias-type Indexmentioning

confidence: 62%

“…We will deduce Theorem 2.1, and hence Corollary 2.4, from an equivariant index theorem for Callias-type operators, Theorem 3.4. This is based on equivariant index theory for such operators with respect to proper actions, developed in [15]. The proof of the index theorem involves several arguments analogous to those in [9].…”

Section: An Index Theoremmentioning

confidence: 99%

“…Let E be the Hilbert C * (G)-module completion of Γ ∞ c (S) with respect to this structure. The definition of the equivariant index of G-Callias-type operators is based on the following result, Theorem 4.19 in [15].…”

Section: An Index Theoremmentioning

confidence: 99%

“…The module E defined above Theorem 3.2 equals E 0 . The following version of the Rellich lemma holds for Sobolev modules (Theorem 3.12 in [15]). We will state and prove a homotopy invariance result, Proposition 4.9, for the index in Definition 3.3, that will be of use later.…”

Section: Properties Of the G-callias-type Indexmentioning

confidence: 99%

“…Proof. This proof is an adaptation of the proof of Theorem 1.14 in [7], with some results from [15] added. For j = 1, 2, 3, 4, let E j and F j be as E and F above and in Theorem 3.2, for the data indexed by j.…”

Section: 4mentioning

confidence: 99%

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Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Guo

2019

Bull. Aust. Math. Soc.

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For a proper action by a locally compact group G on a manifold M with a G-equivariant Spin-structure, we obtain obstructions to the existence of complete G-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where M/G is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in M . We also deduce some other applications of this index theorem. If G is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete G-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

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“…is a well-defined vector in L 2 (S) ⊗ H. By computations similar to those in the proof of Proposition 5.4 in [15], one sees that v L 2 (S)⊗H equals…”

Section: Properties Of the G-callias-type Indexmentioning

confidence: 62%

Section: An Index Theoremmentioning

confidence: 99%

Section: An Index Theoremmentioning

confidence: 99%

Section: Properties Of the G-callias-type Indexmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Guo

2019

Bull. Aust. Math. Soc.

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An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

Hochs

Wang

2022

Math. Z.

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Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) in the K-theory of the reduced group $$C^*$$ C ∗ -algebra $$C^*_rG$$ C r ∗ G of G. This is a common generalisation of the Baum–Connes analytic assembly map and the (equivariant) Atiyah–Patodi–Singer index. In part I of this series, a numerical index $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) was defined for an element $$g \in G$$ g ∈ G , in terms of a parametrix of D and a trace associated to g. An Atiyah–Patodi–Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $$\begin{aligned} \tau _g({{\,\mathrm{index}\,}}_G(D)) = {{\,\mathrm{index}\,}}_g(D), \end{aligned}$$ τ g ( index G ( D ) ) = index g ( D ) , for a trace $$\tau _g$$ τ g defined by the orbital integral over the conjugacy class of g. This implies that the index theorem from part I yields information about the K-theoretic index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) . It also shows that $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) is a homotopy-invariant quantity.

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Higher localised Aˆ-genera for proper actions and applications

Guo

Mathai

2022

Journal of Functional Analysis

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Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

Cited by 11 publications

References 33 publications

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

Higher localised Aˆ-genera for proper actions and applications

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