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.

Asymptotics of complete Kähler metrics of finite volume on quasiprojective manifolds

“…From Theorem 1, we would naively expect the Kähler-Einstein metric to be also an asymptotically tame polyfibred cusp Kähler metric. When dim C X = 1, this is indeed the case as described in [1]. However, when dim C X > 1 and the divisor D is smooth, this is no longer the case as our next result shows (see Theorem 7.4 below for a more precise statement).…”

“…If g ω is an asymptotically tame polyfibred cusp Kähler metric on X and ω t = ω t + √ −1∂∂u(t, ·), with ω t = − Ric(ω) + e −t (ω + Ric(ω)), is the solution to the normalized Ricci flow for t ∈ [0, T ) with ∂u ∂t = log (ω t + √ −1∂∂u) n ω n 0 − u, u(0, ·) = 0, then g ωt is an asymptotically tame polyfibred cusp Kähler metric and u(t, ·) ∈ C ∞ fc ( X) for all t ∈ [0, T ). When dim C X = 1, this result was obtained in [1]. We refer also to [2] for a related result for conformally compact metrics.…”

“…When n = 1, it is more effective to describe the evolution of the metric in terms of a conformal factor, since the evolution equation is then quasilinear instead of fully nonlinear. However, to keep a uniform treatment independent of the dimension of X, we will refrain from doing so and refer to [1] for a study of the spatial asymptotics along the Ricci flow in complex dimension 1 using conformal factors.…”

“…We refer also to [2] for a related result for conformally compact metrics. Our general strategy to prove the result when dim C X > 1 is similar to the one of [1], namely, we restrict the evolution equation of the potential function to D i and solve it to get a candidate u i for what should be the restriction of u to D i . Using a suitable decay estimate proved using a barrier function and the maximum principle (see Proposition 3.6 below), we then check u i is indeed the restriction of u to u i .…”

Ricci flow on surfaces with conic singularities

“…We also mention the work of Rochon [43] where a 'propagation of polyhomogeneity' result is proved in the spirit of Theorem 1.2 but in the complete asymptotically hyperbolic setting, see also Albin-Aldana-Rochon [1], and also the paper by Rochon-Zhang [44] concerning a similar result in higher dimensions.…”