Ricci flows with bursts of unbounded curvature
Ricci flows with bursts of unbounded curvature
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DOI：
10.1080/03605302.2015.1135167
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发表时间：
2016-01-01
影响因子：
1.9
通讯作者：
Topping, Peter M.
中科院分区：
文献类型：
--
作者：
Giesen, Gregor;Topping, Peter M.
给定一个完全任意的曲面，无论它是否具有有界曲率，或者它是否完备，都存在着该曲面在特定时间内存在的瞬时完整的Ricci流演化。在黎曼面支持双曲度量的情况下，这种Ricci流总是存在的，并且收敛到这个双曲度量，即Ricci流几何上升到曲面。在这篇文章中，我们证明了存在完全的，有界曲率的初始度量，包括那些共形于双曲度量的初始度量，它们使得随后的Ricci流在某些中间时刻发展无界曲率。特别地，当与唯一性相结合时，我们发现从这样的初始度量开始的任何完整的Ricci流必然在某个中间时间间隔上发展出无界曲率，但是，尽管存在不涉及无界曲率的双曲度量的其他共形变形，但曲率后来必须是有界的，并且流必须实现几何上升为t。我们构造的另一个结果是，尽管在初始曲面是完整的且曲率有界的特殊情况下，我们的Ricci流最初必须与Hamilton和Shii的经典流一致，但由于唯一性，现在很明显，我们的流通常持续更长的时间间隔，当曲率爆炸时，石氏流停止，但我们的流在这种情况下严格地继续下去。我们所有无界曲率的构造都是在二维发展然后消失的。然后，对更高维度的概括是直接的。
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. 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 ) to this hyperbolic metric, 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 uniqueness, 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, 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 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.