A class of Riemannian manifolds that pinch when evolved by Ricci flow
The author proves that a class of warped product metrics \(g\) -- obtained by warping a positive-Ricci-curvature Einstein manifold \(N\) over the real line \(\mathbb R\) -- will pinch in finite time. The warping is such that the Ricci curvature of \(g\) is nonpositive along \(\mathbb R\) and nonnegative along the fibres \(N\). The paper gives a reasonable exposition of the proof, as well as a brief survey of similar results for the mean curvature flow which inspired the author. In more detail, suppose \(r\) is the diameter of the fibre. The author proves that if \(r\) satisfies certain mild assumptions on its derivatives, then the Ricci flow solution exists, is unique, and is a warped product with the above curvature characteristics for some nonzero time. Bounding the curvature in terms of the derivatives of \(r\) is the key to this, along with guidance from the earlier results of \textit{W.-X. Shi} [J. Differ. Geom. 30, 223-301 (1989; Zbl 0676.53044)]. It follows by a standard argument that the solution exists for a maximal time \(T\leq\infty\), and the curvature must blow up as \(t\to T\) if \(T\) is finite. Pinching happens if \(r\) is unbounded but grows no faster than distance along \(\mathbb R\) -- e.g., if the manifold looks like a cone at either end. Then the author proves that there exists an interval in \(\mathbb R\) on which pinching (i.e. \(r\to 0\)) must occur and to which pinching is confined. It is interesting to contrast this result with the existence of warped product metrics of positive Ricci curvature, constructed using the same ingredients [the reviewer, Proc. Am. Math. Soc. 122, No. 1, 241-246 (1994; Zbl 0812.53045)], for which the flow exists for all time because they are Ricci solitons. One wonders under what initial conditions on the warped product the flow converges to a soliton, rather than pinching; a special case has been considered by the reviewer [Commun. Partial Differ. Equation. 19, 1481-1500 (1994; Zbl 0811.58055)].
scientific article; zbMATH DE number 1406359
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