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

Albin, Pierre; Aldana, Clara Lucia; Rochon, Frédéric

doi:10.1080/03605302.2012.721853

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

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Aldana, Clara Lucia
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Citation

Albin, P., Aldana, C. L., & Rochon, F. (2013). Ricci flow and the determinant of the Laplacian on non-compact surfaces. Communications in partial differential equations, 38 (4): 749, pp. 711. doi:10.1080/03605302.2012.721853.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000E-7CC1-2

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.