Conic degeneration and the determinant of the Laplacian

2012/08/09 by David A. Sher, Sher, David A. · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:58J05 #msc:58J35 #msc:58J50 #msc:58J52

paper · pdf · doi:10.48550/arxiv.1208.1809

41 pages, 7 figures. Version 2: bug fixed in Theorem 2 statement, other minor changes

arxiv created 2013/10/01 · arxiv updated 2013/10/02

Abstract

We investigate the behavior of various spectral invariants, particularly the determinant of the Laplacian, on a family of smooth Riemannian manifolds which undergo conic degeneration; that is, which converge in a particular way to a manifold with a conical singularity. Our main result is an asymptotic formula for the determinant up to terms which vanish as the degeneration parameter goes to zero. The proof proceeds in two parts; we study the fine structure of the heat trace on the degenerating manifolds via a parametrix construction, and then use that fine structure to analyze the zeta function and determinant of the Laplacian.

Cited by

• The determinant on flat conic surfaces with excision of disks

Related

• The determinant on flat conic surfaces with excision of disks
• Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces
• On the spectral geometry of manifolds with conic singularities
• The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points
• Tracking eigenvalues to the frontier of moduli space I: Convergence and spectral accumulation