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Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent

Daniel Grieser, Mohammad Talebi, Boris Vertman

Abstract

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $φ$-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $φ$-manifolds.

Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent

Abstract

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka -) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on -manifolds.

Paper Structure

This paper contains 42 sections, 22 theorems, 226 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction and statement of the main results
  2. Setting of manifolds with fibred boundary metrics
  3. Assumptions and the main result
  4. Key points of the proof
  5. Structure of the paper
  6. Fundamentals of geometric microlocal analysis
  7. Manifolds with corners
  8. Blowup of p-submanifolds
  9. b-vector fields, polyhomogeneous functions and conormal distributions
  10. b-maps and b-fibrations
  11. Operators acting on half-densities
  12. Review of the pseudo-differential b-calculus
  13. b-vector fields and b-differential operators
  14. b-Pseudodifferential operators
  15. Fredholm properties of b-operators
  16. ...and 27 more sections

Key Result

Theorem 1.7

Under the Assumptions assum1, assum2, assum3, assum4 and assum5, the Schwartz kernel of the resolvent $(\Delta_{\phi} + k^{2})^{-1}, k>0,$ lifts to a polyhomogeneous conormal distribution on an appropriate manifold with corners, with a conormal singularity along the diagonal.

Figures (16)

  • Figure 1: b-double space $M^{2}_{\mathrm{b}}$ and $\beta_{\mathrm{b}}: M^{2}_{\mathrm{b}} \to {\overline{M}}^2$.
  • Figure 2: $\phi$-double space $M_{\phi}^{2}$
  • Figure 3: Schematic structure of index sets of $Q_\alpha$
  • Figure 4: Blowup of $C_{11}^\bullet$
  • Figure 5: Blow up of $C_{01}^\bullet, C_{10}^\bullet, C_{11}^+.$
  • ...and 11 more figures

Theorems & Definitions (58)

  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Conormal distributions
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 48 more