Uniqueness and nonuniqueness for Ricci flow on surfaces: reverse cusp singularities
type: Item
instance of: scholarly article
zbMATH Open document ID: 1246.53092
publication date: 2012-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.2795
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 6043646
zbMATH Keywords: Ricci flow
zbMATH Keywords: cusp singularities
zbMATH Keywords: nonuniqueness
zbMATH Keywords: initial condition
zbMATH Keywords: flows contracting cusp
author: Peter M. Topping
MaRDI profile type: Publication
OpenAlex ID: W2963079691
arXiv ID: 1010.2795
DOI: 10.1093/IMRN/RNR082
published in: IMRN. International Mathematics Research Notices
title: Uniqueness and nonuniqueness for Ricci flow on surfaces: reverse cusp singularities
review text: The author introduces a notion of initial condition for a complete Ricci flow on a manifold \(\mathcal{M}\), \(\frac{\partial}{\partial t} g(t)=-2\mathrm{Ric}[g(t)]\), for \(t\in (0,T]\), as a complete Riemannian manifold \((\mathcal{N},g_0)\) satisfying the condition \(\varphi^*(g(t))\to g_0\) smoothly locally on \(\mathcal{N}\) when \(t \downarrow 0\), for some smooth map \(\varphi:\mathcal{N}\to \mathcal{M}\) that is a diffeomorphism onto \(\varphi(\mathcal{N})\). This generalizes the usual notion of initial metric when \(\mathcal{N}= \mathcal{M}\), \(\varphi\) is the identity map, and \(g(0)=g_0\).NEWLINENEWLINEUnder this kind of initial condition, existence and uniqueness of the Ricci flow is discussed. The author gives as an example of \(\mathcal{M}\) the 2-torus and \(\mathcal{N} \subset \mathcal{M}\) a once-punctured 2-torus with the conformal hyperbolic metric, allowing a Ricci flow continuation covering the whole torus \(\mathcal{M}\), whose cusp contracts logarithmically, and so different from the known case of homothetically expanding keeping the cusp. This flow changes the initial topological type adding the removed point at infinity. A general result is given in Theorem 1.2 for compact Riemann surfaces \(\mathcal{M}\) and \(\mathcal{N}= \mathcal{M}\backslash \{p_1, \dots, p_n\}\) with any conformal metric \(g_0\) of bounded-curvature and negative curvature on the neighbourhood of each point \(p_i\). A complete Ricci flow \(g(t)\) with curvature uniformly bounded from below as \(t\downarrow 0\) can be constructed with initial condition \((\mathcal{N}, g_0)\) and satisfying \(\frac{1}{C}(-\ln t)\leq \mathrm{diam}(\mathcal{M}, g(t))\leq C(-\ln t)\), for some constant \(C<\infty\), and \(t\) sufficiently small.NEWLINENEWLINEThe existence of this flow is obtained by constructing an increasing sequence of approximating conformal metrics \(g_k\) to \(g_0\) on punctured discs, with suitable behaviour of the corresponding Ricci flows. The upper estimate of the …
reviewed by: Isabel Salavessa
description: scientific article; zbMATH DE number 6043646