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Resolvent, heat kernel, and torsion under degeneration to fibered cusps
About this Title
Pierre Albin, Frédéric Rochon and David Sher
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 269, Number 1314
ISBNs: 978-1-4704-4422-8 (print); 978-1-4704-6466-0 (online)
DOI: https://doi.org/10.1090/memo/1314
Published electronically: March 23, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Fibered cusp surgery metrics
- Resolvent under degeneration
- 3. Pseudodifferential operator calculi
- 4. Resolvent construction
- 5. Projection onto the eigenspace of small eigenvalues
- Heat kernel under degeneration
- 6. Surgery heat space
- 7. Solving the heat equation
- Torsion under degeneration
- 8. The $R$-torsion on manifolds with boundary
- 9. The intersection $R$-torsion of Dar and $L^2$-cohomology
- 10. Analytic torsion conventions
- 11. Asymptotics of analytic torsion
- 12. A Cheeger-Müller theorem for fibered cusp manifolds
- A. Model cases: Euclidean Laplacians and Dirac operators
- B. Geometric microlocal preliminaries
- C. Proof of composition formula
Abstract
Manifolds with fibered cusps are a class of complete non-compact Riemannian manifolds including many examples of locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.- Pierre Albin, Clara L. Aldana, and Frédéric Rochon, Ricci flow and the determinant of the Laplacian on non-compact surfaces, Comm. Partial Differential Equations 38 (2013), no. 4, 711–749. MR 3040681, DOI 10.1080/03605302.2012.721853
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