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arxiv: 2402.02398 · v4 · submitted 2024-02-04 · 🧮 math.DG

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

Pengshuai Shi This is my paper

Pith reviewed 2026-05-24 03:42 UTC · model grok-4.3

classification 🧮 math.DG
keywords relative eta invariantDirac-Schrödinger operatorsuniformly positive scalar curvaturenon-compact manifoldsspectral flowAtiyah-Patodi-Singer theoremGromov-Lawson resultindex theorem
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The pith

Relative eta invariant provides spectral flow formula and generalizes Atiyah-Patodi-Singer theorem to non-compact boundaries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the relative eta invariant for Dirac-Schrödinger operators that coincide at infinity on complete non-compact manifolds with bounded sectional curvature. It applies this to spin Dirac operators on manifolds admitting uniformly positive scalar curvature metrics, producing a formula for spectral flow that gives a new proof of the Gromov-Lawson result on area enlargeable manifolds in odd dimensions. The work also derives an index formula for operators on manifolds with non-compact boundaries, extending the Atiyah-Patodi-Singer theorem, and uses the invariant to examine the space of positive scalar curvature metrics on certain non-compact connected sums.

Core claim

On complete non-compact manifolds with bounded sectional curvature, the relative eta invariant of Dirac-Schrödinger operators that coincide at infinity provides a geometric expression for spectral flow and an index formula when the operators arise as boundary restrictions on manifolds with non-compact boundary, generalizing the Atiyah-Patodi-Singer theorem and enabling study of uniformly positive scalar curvature metrics.

What carries the argument

Relative eta invariant for pairs of Dirac-Schrödinger operators that coincide at infinity, which encodes the difference in spectral data and allows index and flow calculations without compactness.

If this is right

  • The spectral flow between two such operators on non-compact manifolds equals a geometric quantity derived from the relative eta invariant.
  • A new proof exists for the non-existence of uniformly positive scalar curvature metrics on certain compact area enlargeable manifolds in odd dimensions.
  • The Atiyah-Patodi-Singer index theorem holds when the boundary is non-compact under the given conditions.
  • The relative eta invariant distinguishes components in the space of uniformly positive scalar curvature metrics on non-compact connected sums.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the relative eta invariant vanishes, it may imply that two metrics of positive scalar curvature are connected by a path preserving the property on non-compact manifolds.
  • This approach could extend to other elliptic operators beyond Dirac type on non-compact settings.
  • Applications might include obstructions to positive scalar curvature on more general non-compact manifolds constructed via connected sums.

Load-bearing premise

The two Dirac-Schrödinger operators must coincide at infinity on complete non-compact manifolds with bounded sectional curvature to allow definition of the relative eta invariant.

What would settle it

A pair of Dirac-Schrödinger operators coinciding at infinity on a manifold with bounded sectional curvature where the computed index using the relative eta invariant differs from the actual Fredholm index of the associated operator.

read the original abstract

On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schr\"odinger operators. Assuming two Dirac-Schr\"odinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac-Schr\"odinger 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops the theory of relative eta invariants for Dirac-Schrödinger operators on complete non-compact manifolds with bounded sectional curvature where the operators coincide at infinity. It derives a geometric formula for spectral flow, provides a new proof of Gromov-Lawson's result for compact area-enlargeable manifolds in odd dimensions, generalizes the Atiyah-Patodi-Singer index theorem to manifolds with non-compact boundary, and applies the results to study uniformly positive scalar curvature metrics on certain non-compact connected sums.

Significance. If the foundational constructions hold under the stated hypotheses, the work extends index theory and eta invariants to non-compact settings and supplies new tools for positive scalar curvature questions on non-compact manifolds. The new proof of Gromov-Lawson and the generalized index formula would be notable contributions.

major comments (1)
  1. [Abstract] Abstract and setup: the relative eta invariant is defined via previous work under the sole hypotheses of completeness, non-compactness, and bounded sectional curvature. The manuscript must explicitly confirm (with a precise citation to the relevant theorem or proposition in the cited prior work) that these hypotheses suffice, or state any additional decay/regularity conditions required on the difference of the operators or on the curvature. This verification is load-bearing for the spectral-flow formula, the Gromov-Lawson proof, the non-compact-boundary index formula, and the PSC applications.
minor comments (2)
  1. Add explicit cross-references to the precise statements in the cited previous work when invoking the definition of the relative eta invariant.
  2. Clarify the precise meaning of 'coincide at infinity' with an equation or asymptotic statement in the setup section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and setup: the relative eta invariant is defined via previous work under the sole hypotheses of completeness, non-compactness, and bounded sectional curvature. The manuscript must explicitly confirm (with a precise citation to the relevant theorem or proposition in the cited prior work) that these hypotheses suffice, or state any additional decay/regularity conditions required on the difference of the operators or on the curvature. This verification is load-bearing for the spectral-flow formula, the Gromov-Lawson proof, the non-compact-boundary index formula, and the PSC applications.

    Authors: We agree that an explicit confirmation with a precise citation is required. In the revised manuscript we will insert, immediately after the sentence defining the relative eta invariant in the introduction (and a corresponding clarifying clause in the abstract), the following statement: 'By [precise citation to the authors' prior paper], Theorem X (or Proposition Y), the relative eta invariant is well-defined for any pair of Dirac-Schrödinger operators on a complete non-compact manifold with bounded sectional curvature that coincide at infinity; no further decay or regularity assumptions on the difference of the operators or on the curvature are needed beyond the stated hypotheses.' This citation is to the exact result in the prior work that establishes the construction under precisely these conditions. The revision will be made in Section 1 and the abstract. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior definition of relative eta; central applications and new proof remain independent of internal fits or redefinitions.

full rationale

The paper attributes the definition of the relative eta invariant directly to 'previous work' for Dirac-Schrödinger operators coinciding at infinity (abstract). All subsequent results—the geometric spectral-flow formula, new proof of Gromov-Lawson for area-enlargeable manifolds, APS-type index formula on non-compact boundaries, and PSC-metric applications—are presented as consequences of this externally defined invariant under the stated hypotheses of completeness and bounded sectional curvature. No equations inside the paper redefine the eta invariant in terms of its own outputs, fit parameters to data then rename them predictions, or smuggle an ansatz via self-citation. The derivation chain therefore does not reduce by construction to its own inputs; the cited prior construction supplies independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions of differential geometry rather than introducing new free parameters or invented entities; all listed items are background facts required for the setting.

axioms (2)
  • domain assumption Manifolds are complete, non-compact, and have bounded sectional curvature
    Explicitly stated as the setting in which Dirac-Schrödinger operators are considered and the relative eta invariant is defined.
  • domain assumption The two Dirac-Schrödinger operators coincide at infinity
    Required condition for the definition of the relative eta invariant.

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Reference graph

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