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.
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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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DOI: https://doi.org/10.1007/978-3-642-16286-2_10
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-16286-2
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