Ricci Solitons
The art of doing mathematics consists in finding that special case which con-
tains all the germs of generality. - David Hilbert
The last thing one knows when writing a book is what to put first.
- Blaise Pascal
The notion of soliton solutions to evolution equations first appeared in
connection with the modelling of shallow water waves and the Korteweg-
de Vries equation [131]. In this context, a soliton is, in the memorable
phrase of Scott-Russell, a 'wave of translation', i.e., a solitary water wave
moving by translation, without losing its shape. More generally, we now
think of solitons as self-similar solutions, i.e., solutions which evolve along
symmetries of the flow. In the case of the Ricci flow, these symmetries are
scalings and diffeomorphisms.
In this chapter we study Ricci solitons and in particular the special case
of gradient Ricci solitons. In later chapters we shall further see how these
special solutions of the Ricci flow motivate the general analysis of the Ricci
flow through monotonicity formulas and their subsequent applications. In
particular, Ricci solitons have inspired the entropy and Harnack estimates,
the space-time formulation of Ricci flow, and the reduced distance and re-
duced volume. Furthermore, the entropy and reduced volume monotonicity
formulas have the geometric application of no local collapsing, which is fun-
damental in the study of singularities (see the diagram below). Gradient
Ricci solitons also model the high curvature regions of singular solutions.
This motivates trying to classify gradient Ricci solitons, especially in low
Gradient Ricci solitons
/ /
Entropy Harnack Space-time Red. dist. & vol.
No local collapsing
Red. dist. &; vol." is an abbreviation for reduced distance and volume, to be
discussed in Chapter 7.
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