Uniqueness and nonuniqueness for Ricci flow on surfaces: reverse cusp singularities
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 decay of the diameter of \((\mathcal{M}, g(t))\) is derived considering estimates of the distance function for the Ricci flow of the approximated metrics, and this is achieved by constructing suitable barrier functions on punctured discs \(D\). The lower estimate is obtained by direct inspection of the Gauss curvature on \(\partial D_r\), for disks of sufficiently small radius \(r\). In spite the of nonuniqueness of the flow, it is shown in Theorem 1.5 that is unique among all these kinds of flows contracting cusps and with Gauss curvature uniformly bounded from below. It is explained that the curvature of \(g(t)\) blows up when \(t\downarrow 0\) with a type II(c) singularity. Theorem 1.2 also answers positively to a question of Perelman on the necessity of the condition \(Vol(B(x_0, r_0))\geq (1-\delta)\omega_nr_0^n\) on the volume ratio in his pseudolocality theorem, where \(n=\mathrm{dim}\mathcal{M}\) [\textit{G. Perelman}, ``The entropy formula for the Ricci flow and its geometric applications'', arXiv e-print service, Cornell University Library (2002; Zbl 1130.53001)].NEWLINENEWLINEFurthermore, in Theorem 1.3, by using curvature estimates of the Ricci flow along geodesic balls (proved in detail in [\textit{B.-L. Chen}, J. Differ. Geom. 82, No. 2, 363--382 (2009; Zbl 1177.53036)]), it is shown that if the volume of all geodesic balls of some radius \(r_0\) of the initial Riemannian manifold \((\mathcal{N}, g_0)\) has a positive lower bound, then \((\mathcal{M},g(t))\) has uniformly bounded curvature for \(t\) in some interval \( [0, \delta]\), \(\varphi(\mathcal{N})=\mathcal M\), and \(g(t)\) is smoothly extended to \(t=0\) as \(\varphi_*g_0\). Thus, with such condition on the initial Riemannian manifold \((\mathcal{N},g_0)\) uniqueness of the Ricci flow holds up to a diffeomorphism.
scientific article; zbMATH DE number 6043646