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Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds

Pengshuai Shi

TL;DR

The paper develops a relative eta- invariant framework for Dirac–Schrödinger operators on complete non-compact manifolds with bounded geometry and shows a real gluing formula in the invertible case. It derives a geometric spectral-flow formula linking the flow of twisted Dirac operators to the integral of the A-hat genus against Chern–Simons forms, and establishes APS-type index theorems for manifolds with non-compact boundary. These tools are then applied to study spaces of uniformly positive scalar curvature metrics on non-compact connected sums, proving nontrivial topology (infinite path components) of PSC metric spaces and distinguishing metrics via relative rho invariants. The results extend index theory and PSC-geometry to non-compact settings, offering new methods to understand moduli spaces of PSC metrics and the impact of non-compact boundaries on index-theoretic invariants.

Abstract

On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schrödinger operators. Assuming two Dirac-Schrödinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac-Schrödinger operators is the (twisted) spin Dirac operators on spin manifolds which admit a Riemannian metric of uniformly positive scalar curvature. In this case, using the relative eta invariant, we get a geometric formula for the spectral flow on non-compact manifolds, which induces a new proof of Gromov-Lawson's result about compact area enlargeable manifolds in odd dimensions. When two such spin Dirac operators are the boundary restriction of an operator on a manifold with non-compact boundary, under certain conditions, we obtain an index formula involving the relative eta invariant. This generalizes the Atiyah-Patodi-Singer index theorem to non-compact boundary situation. As a result, we can use the relative eta invariant to study the space of uniformly positive scalar curvature metrics on some non-compact connected sums.

Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds

TL;DR

The paper develops a relative eta- invariant framework for Dirac–Schrödinger operators on complete non-compact manifolds with bounded geometry and shows a real gluing formula in the invertible case. It derives a geometric spectral-flow formula linking the flow of twisted Dirac operators to the integral of the A-hat genus against Chern–Simons forms, and establishes APS-type index theorems for manifolds with non-compact boundary. These tools are then applied to study spaces of uniformly positive scalar curvature metrics on non-compact connected sums, proving nontrivial topology (infinite path components) of PSC metric spaces and distinguishing metrics via relative rho invariants. The results extend index theory and PSC-geometry to non-compact settings, offering new methods to understand moduli spaces of PSC metrics and the impact of non-compact boundaries on index-theoretic invariants.

Abstract

On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schrödinger operators. Assuming two Dirac-Schrödinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac-Schrödinger operators is the (twisted) spin Dirac operators on spin manifolds which admit a Riemannian metric of uniformly positive scalar curvature. In this case, using the relative eta invariant, we get a geometric formula for the spectral flow on non-compact manifolds, which induces a new proof of Gromov-Lawson's result about compact area enlargeable manifolds in odd dimensions. When two such spin Dirac operators are the boundary restriction of an operator on a manifold with non-compact boundary, under certain conditions, we obtain an index formula involving the relative eta invariant. This generalizes the Atiyah-Patodi-Singer index theorem to non-compact boundary situation. As a result, we can use the relative eta invariant to study the space of uniformly positive scalar curvature metrics on some non-compact connected sums.
Paper Structure (23 sections, 27 theorems, 108 equations)

This paper contains 23 sections, 27 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Summary of the main results
  3. Organization of this paper
  4. Relative eta invariant and a gluing formula in the invertible case
  5. Dirac-type operators
  6. The relative eta invariant
  7. Relative eta invariant on manifolds with boundary
  8. Proof of the gluing formula
  9. The spectral flow on non-compact manifolds
  10. Variation of the truncated relative eta invariants
  11. The spectral flow formula
  12. Chern--Simons forms and the spectral flow
  13. Area enlargeable manifolds
  14. Index theorem related to manifolds with uniformly PSC metrics
  15. Space of uniformly PSC metrics which coincide at infinity
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose that $M$ is an odd-dimensional non-compact spin manifold of bounded sectional curvature which admits a metric $g$ of uniformly PSC. For $r\in[0,1]$, let $\nabla_r^F$ be a linear path of connections connecting two flat connections $\nabla_0^F$ and $\nabla_1^F$ on a Hermitian vector bundle $F$ where $\hat{A}(M,g)$ is the $\hat{A}$-genus form of $(M,g)$, and $\operatorname{Tch}(\nabla_0^F,\na

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5: Shi22
  • Definition 2.6
  • Remark 2.8
  • ...and 52 more