A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In this survey, we discuss the global behavior of the solutions to these equations. In particular, we describe how these techniques can be used to prove the Differentiable Sphere Theorem.

中文翻译：

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黎曼几何中的演化方程

黎曼几何的一个基本问题是在给定的光滑流形上找到规范度量。在1980年代，RS Hamilton提出了一种基于抛物型偏微分方程的方法。目标是从给定的初始度量开始，然后通过演化方程将其变形为规范度量。黎曼度量有多种自然演化方程，包括Ricci流和保形Yamabe流。在本次调查中，我们讨论了这些方程解的整体行为。特别是，我们描述了如何使用这些技术来证明微分球定理。