具有无界曲率爆发的里奇流
Ricci flows with bursts of unbounded curvature
摘要：
给定一个完全任意的曲面，无论它是否具有有界曲率，甚至无论它是否是完整的，都存在该曲面的瞬时完整里奇流演化，该演化存在特定的时间量[GT 11]。在基础黎曼曲面支持双曲度规的情况下，这个里奇流总是存在于所有时间，并收敛（在缩放因子$\frac1{2t}$之后）到这个双曲度规[GT 11]，即我们的里奇流几何化了曲面。在本文中，我们表明，存在完整的，有界曲率的初始度量，包括那些共形的双曲度量，随后里奇流开发无界曲率在某些中间时间。特别是，当与[Top13]的唯一性相结合时，我们发现任何以这样的初始度量开始的完整里奇流必须在某个中间时间间隔上发展出无界曲率，但是尽管如此，曲率后来必须成为有界的，并且流必须实现几何化为$t\to\infty$，即使双曲度量有其他不涉及无界曲率的共形变形。我们的构造的另一个结果是，虽然我们的里奇流从[GT 11]必须同意最初与汉密尔顿和施的经典流的特殊情况下，初始表面是完整的和有界曲率，通过唯一性，现在很明显，我们的流持续了一个更长的时间间隔一般，与施的流停止时，曲率爆炸，但我们的流动在这些情况下严格地继续下去。我们所有关于无限曲率发展然后消失的构造都是二维的。于是，更高维度的普遍化是立即的。
英文摘要：
Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor $\frac1{2t}$) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as $t\to\infty$, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature. Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi's flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations. All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate.
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