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

TL;DR

This paper extends the analysis of the Laplacian determinant's maximization from compact to non-compact surfaces, demonstrating that Ricci flow leads to metrics of constant curvature while increasing the determinant.

Contribution

It generalizes the known results for compact surfaces to non-compact surfaces with hyperbolic ends, showing Ricci flow convergence and determinant increase.

Findings

Ricci flow converges to constant curvature metrics on non-compact surfaces.

The determinant of the Laplacian increases along the Ricci flow.

Extension of compact surface results to non-compact geometries.

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.