1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

CONTENTS OF PART I OF VOLUME TWO xxi this energy. The nonexistence of steady and expanding breathers on closed manifolds, origin ...

xxii CONTENTS OF PART I OF VOLUME TWO the Ricci flow. We also consider the reduced distance in the special cases of Einstein sol ...

CONTENTS OF PART I OF VOLUME TWO xxiii classes of complete solutions with bounded curvature on noncompact mani- folds diffeomorp ...

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CHAPTER 1 Ricci Solitons The art of doing mathematics consists in finding that special case which con- tains all the germs of ge ...

2 1. RICCI SOLITONS Some highlights of this chapter are a systematic description of the equa- tions obtained from differentiatin ...

GENERAL SOLITONS AND THEIR CANONICAL FORMS 3 By choosing O" (t) so that 0-(t) changes sign, this soliton may both expand and s ...

4 1. RICCI SOLITONS A way to circumvent the undesirableness of the uniqueness assumption in Proposition 1.3 is as follows. Since ...

GENERAL SOLITONS AND THEIR CANONICAL FORMS 5 (1) cp(t) : M --t M is the 1-parameter family of diffeomorphisms gen- erated by X ...

6 1. RICCI SOLITONS is the exponent in the Gaussian function. Here we used the standard identity £vfgcan = 2'\7'\7 f. 2. Differe ...

DIFFERENTIATING THE SOLITON EQUATION 7 PROOF. The left-hand side (LHS) is just the usual expression for ~~ under the Ricci fl. ...

8 1. RICCI SOLITONS Now we proceed to systematically differentiate the function f up to fourth order, commute pairs of covariant ...

DIFFERENTIATING THE SOLITON EQUATION 9 In conclusion, the new identities we obtain by differentiating f up to fourth order are ...

10 1. RICCI SOLITONS COROLLARY 1.16. If (g, \7 f) is a steady gradient Ricci soliton structure on Mn with positive Ricci curvatu ...

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 11 Sections 4.3 and 5 of Chapter 3 of [108]). Once this condition is added to (1.17 ...

12 1. RICCI SOLITONS PROOF. Let ( M^2 , g) be a Riemannian surface (not necessarily complete) with a Killing vector field K. Let ...

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 13 where r is the standard coordinate on an interval I c ~' g is a given metric on ...

14 1. RICCI SOLITONS where N = 81 , and it is natural to assume that Rc(g) + \7^2 f = 0 for a radial function f(r). Using (1.38) ...

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 15 Likewise, taking b = 0, we have the ODE w' = aw^2. If a = -a^2 , then w (r) = °' ...

16 1. RICCI SOLITONS (1) a constant negative curvature metric on a surface transforming to a steady soliton, (2) an incomplete s ...