Resolvent, heat kernel, and torsion under degeneration to fibered cusps

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.Contents 1. Introduction 1 2. Fibered cusp surgery metrics 8 Resolvent under degeneration 15 3. Pseudodifferential operator calculi 15 4. Resolvent construction 21 5. Projection onto the eigenspace of small eigenvalues 40 Heat kernel under degeneration 43 6.Surgery heat space 43 7. Solving the heat equation 48 Torsion under degeneration 64 8.The R-torsion on manifolds with boundary 64 9.The intersection R-torsion of Dar and L 2 -cohomology 68 10.Analytic torsion conventions 72 11.Asymptotics of analytic torsion 77 12.A Cheeger-Müller theorem for fibered cusp manifolds 83 Appendix A. Model cases: Euclidean Laplacians and Dirac operators 85 Appendix B. Geometric microlocal preliminaries 89 Appendix C. Proof of composition formula 92

Locations

• Memoirs of the American Mathematical Society - View
• arXiv (Cornell University) - View - PDF