1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

CONSTRUCTING THE BRYANT STEADY SOLITON 17 We have the following from Theorem 13 on p. 61 and Theorem 10 on p. 55 of Huber [210 ...

18 1. RICCI SOLITONS 4.1. Setting up the ODE for Bryant solitons. With p = n - 1, substituting (1.38) and (1.39) into the steady ...

CONSTRUCTING THE BRYANT STEADY SOLITON FIGURE 1. Phase portrait for system (l.48), n = 2, drawn using Maple. Since :ft log w = ...

20 1. RJCCI SOLITONS saddle points since the linearization of (1.48) at (1, n) is^8 du dt = - (n - 2) u - v, dv = -nu+v dt ' an ...

CONSTRUCTING THE BRYANT STEADY SOLITON 21 the differential equation for y in (1.48) shows that there exists to such that y ::: ...

22 1. RICCI SOLITONS new phase plane coordinates x X~y'n-, y y ~ y'n(n - 1). y As x -----t O+ and y -----t +oo, X -----t O+ and ...

CONSTRUCTING THE BRYANT STEADY SOLITON 23 for some c between a and b. Letting b-+ +oo gives g'(c)/h'(c) = g(a)/h(a). Substitut ...

24 1. RICCI SOLITONS Now we derive the asymptotic behavior of wand r. We have d x dY (log w + log Y) = y ( X _ a) , which by Lem ...

and (1.57) CONSTRUCTING THE BRYANT STEADY SOLITON x^2 -xy+n-1 w"=------ w (n -1) (X^2 -yin (X - CYY^2 )) wY^2 = O(-Y^3 ) = O(- ...

26 1. RICCI SOLITONS is positive everywhere else. The curvatures approach zero as r -----t +oo, so R has a positive maximum valu ...

ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 27 on JRn+l, we get a system of two second-order ODES for f and w as functions of r: ...

RICCI SOLITONS interested in solutions in the half-space where. w is positive. There is a 1- parameter family of these tra ...

ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 29 This establishes that X < a always along these trajectories, and the equa- tio ...

30 1. RICCI SOLITONS PROOF. In these coordinates, the sectional curvatures are 1-(w')^2 Y^2 - (n -.l)X^2 (X^2 + Y^2 + n>.W^2 ...

ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 31 Hence W^2 / X 2::: c1 for some positive c1, for r sufficiently large. Thus, the d ...

32 1. RICCI SOLITONS for some constant C > 0. Because dr = Wds and W = 0(1/r), then d w^2 -log-< cr-^3 dr Y - ' from which ...

HOMOGENEOUS EXPANDING SOLITONS 33 Many noncompact homogeneous spaces admit no Einstein metrics what- soever. For instance, the ...

34 1. RICCI SOLITONS One begins with the observation that conditions (1.64)-(1.66) are equiv- alent for a nilpotent group. LEMMA ...

HOMOGENEOUS EXPANDING SOLITONS 35 Lauret also gives a variational characterization of Ricci solitons [244]. Roughly speaking, ...

36 1. RICCI SOLITONS 6.2. Construction. The first explicit examples of Ricci soliton struc- tures on Lie groups were constructed ...