CHAPTER 1 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 dimensions.1 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. 1
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