Ricci flow and the determinant of the Laplacian on non-compact surfaces

Abstract

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow converges to a metric of constant curvature and that the determinant increases along this flow.

40 pages, 4 figures, other minor changes

References in corpus (5)

• Existence of Ricci flows of incomplete surfaces
• Index and Spectral Theory for Manifolds with Generalized Fibred Cusps
• Ricci flow of conformally compact metrics
• Normalized Ricci flow on Riemann surfaces and determinants of Laplacian
• A local families index formula for d-bar operators on punctured Riemann surfaces

Cited by in corpus (5)

• Ricci flow on surfaces with conic singularities
• Uniqueness of Instantaneously Complete Ricci flows
• A Polyakov formula for sectors
• Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula
• Polyakov formulas for conical singularities in two dimensions