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How to produce a Ricci flow via Cheeger-Gromoll exhaustion. (English) Zbl 1351.53078

The authors consider the Ricci flow \[ \frac{\partial }{\partial t} g(t)=-2 \text{Ric }(g(t))\eqno{(*)} \] on open (i.e., complete and non-compact) \(n\)-dimensional manifolds with non-negative complex sectional curvature \(K^{\mathbb{C}}\geq 0\).
The short-time existence of the flow \((*)\) was settled for closed manifolds by R. S. Hamilton [J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)] and for non-compact \(2\)-dimensional manifolds (possibly incomplete) by G. Giesen and P. M. Topping [Commun. Partial Differ. Equations 36, No. 10–12, 1860–1880 (2011; Zbl 1233.35123)]. The case of non-compact manifolds of higher dimension \(n\geq 3\) seems to be very hard to attack without any curvature condition.
The main result of the paper is the short-time existence of the flow \((*)\) on open manifolds with complex sectional curvature \(K^{\mathbb{C}}\geq 0\). The latter condition was previously known to be preserved under the Ricci flow [S. Brendle and R. Schoen, J. Am. Math. Soc. 22, No. 1, 287–307 (2009; Zbl 1251.53021); L. Ni and J. Wolfson, “Positive complex sectional curvature, Ricci flow and the differential sphere theorem”, Preprint, arXiv:0706.0332].
More precisely, the authors prove that given an open manifold \((M, g)\) with non-negative (and possibly unbounded) complex sectional curvature, there exists a time \(\mathcal{T}\) such that \((*)\) has a smooth solution \(g(t)\) in \([0, \mathcal{T}]\) with \(g(0)=g\). Moreover, they give a precise lower bound on the existence time \(\mathcal{T}\) of the flow in terms of the supremum of volume of balls.
As a corollary, they can insure the existence of an immortal solution (\(\mathcal{T}=+\infty\)) if the manifold \(M\) has volume growth faster than \(r^{n-2}\). This volume growth condition is also optimal. Long-time existence was previously known only in the case of Euclidean volume growth and under stronger assumptions [F. Schulze and M. Simon, Math. Z. 275, No. 1–2, 625–639 (2013; Zbl 1278.53072)].
The proof of the main result is easier in the case \(K_g^\mathbb{C}>0 \). This is because of the existence of a smooth strictly convex proper function \(\beta: M\rightarrow [0,+\infty)\) that in particular has a global minimum point \(p_0\). The idea is to look at the double of the compact and convex sublevel sets \(C_i:=\beta^{-1}([0,i])\), denoted by \(M_i\), and to show that it is a smooth closed manifold that admits a metric \(g_i\) such that \(K_{g_i}^\mathbb{C}\geq 0\). The pointed sequence \((M_i, g_i, p_0)\) then converges (in the sense of smooth Cheeger-Gromov convergence) to \((M, g)\). By [R. S. Hamilton, J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)], one can run the Ricci flow on \((M_i, g_i, p_0)\) until the maximal time of existence \(T_i\). One of the key points is to show that there is a uniform (i.e., independent of \(i\)) lower bound for \(T_i\), say \(\mathcal{T}\). The short-time existence for the positively curved case then follows proving that the sequence of closed Ricci flow \((M_i, g_i(t), p_0)\) converges to a complete limit solution.
Some additional difficulties arise when \(K_g^\mathbb{C}\geq 0 \). In the paper the authors deal with several technical results in order to circumvent these issues and to be able to try and adapt the same idea used in the case \(K_g^\mathbb{C} > 0 \).
As mentioned above, the short-time existence result holds for an initial metric \(g\) with \(K_g^\mathbb{C}\geq 0\) and with possibly unbounded curvature. Under a non-collapsing condition, the authors are also able to prove that the curvature of the evolving manifold under the Ricci flow \((M, g(t))\) is bounded above for any positive time \(t\in(0, \mathcal{T})\). The non-collapsing assumption is always satisfied in dimension \(2\) while such a condition is shown to be essential in the case \(n\geq 3\).

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K45 Initial value problems for second-order parabolic systems
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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