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The \(\mathcal{F}\)-Functional and Gradient Flows

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2011)

Abstract

After Ricci flow was first introduced, it appeared for many years that there was no variational characterisation of the flow as the gradient flow of a geometric quantity. In particular, Bryant and Hamilton established that the Ricci flow is not the gradient flow of any functional on Met – the space of smooth Riemannian metrics – with respect to the natural L2 inner product (with the exception of the two-dimensional case, where there is indeed such an ‘energy’). Considering the prominent role variational methods have played in geometric analysis, pde’s and mathematical physics, it seemed surprising that such a natural equation as Ricci flow should be an exception. One of the many important contributions Perel’man made was to elucidate a gradient flow structure for the Ricci flow, not on Met but on a larger augmented space. Part of this structure was already implicit in the physics literature [Fri85]. In this chapter we discuss this structure, at the centre of which is Perel’man’s F-functional [Per02]. The analysis will provide the ground work for the proof of a lower bound on injectivity radius at the end of Chap. 11.

Keywords

  • Riemannian Metrics
  • Gradient Flow
  • Ricci Flow
  • Natural Equation
  • Injectivity Radius

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Correspondence to Ben Andrews .

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© 2011 Springer-Verlag Berlin Heidelberg

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Andrews, B., Hopper, C. (2011). The \(\mathcal{F}\)-Functional and Gradient Flows. In: The Ricci Flow in Riemannian Geometry. Lecture Notes in Mathematics(), vol 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16286-2_10

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