Resolvent, heat kernel and torsion under degeneration to fibered cusps

2014/10/30 by Albin, Pierre, Rochon, Frédéric, Sher, David
#55N25 #55N33 #58J05 #58J35 #58J50 #58J52 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1410.8406

Abstract

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all 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.

Related

• Analysis on fibred cusp spaces
• Analytic torsion on manifolds with fibred boundary metrics
• Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics
• Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion
• A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics