1547671870-The_Ricci_Flow__Chow

THE RICCI FLOW OF HOMOGENEOUS GEOMETRIES 9 metric g.. Then one may define a vector space isomorphism g --+ /\^2 g by Composing ...

10 1. THE RICCI FLOW OF SPECIAL GEOMETRIES If we want to evolve g by the Ricci flow, we must study its curvature. For brevity, w ...

A GEOMETRY WITH ISOTROPY SO (3) 11 By the lemma, any choice of Milnor frame for a left-invariant metric g on g lets us globall ...

i2 1. THE RICCI FLOW OF SPECIAL GEOMETRIES curvature by shrinking the circles of the Hopf fibration. The Gromov- Hausdorff limit ...

A GEOMETRY WITH ISOTROPY SO (3) 13 From the symmetry in these equations, we may assume without loss of generality that Do :::; ...

14 1. THE RICCI FLOW OF SPECIAL GEOMETRIES LEMMA 1.19. There exists a time To > 0 depending only on the initial data (Ao, Bo, ...

A GEOMETRY WITH ISOTROPY SO (2) there is by (1.6) some k3 < oo such that !!:F <PF= ±_AF< k^3 dt - BC - k2 (k1 - 8t). ...

16 1. THE RICCI FLOW OF SPECIAL GEOMETRIES of upper-triangular 3 x 3 matrices g' ~ { G ~ 0 x, y, z E ~} endowed with the usual m ...

A GEOMETRY WITH TRIVIAL ISOTROPY 17 Observing that AB is another conserved quantity, we can thus obtain the full solution (1.8 ...

18 1. THE RICCI FLOW OF SPECIAL GEOMETRIES so the Ricci flow is equivalent to the system (1.lOa) d C^2 - A^2 dtA =^4 BC (1.lOb) ...

NOTES AND COMMENTARY 19 where a and /3 are constant and 'Y is a linear function of e such that ( ~ e-°' ( d-y /dB) > 0. It is ...

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CHAPTER 2 Special and limit solutions In this chapter, we continue our study of special and intuitive solutions to the Ricci flo ...

22 2. SPECIAL AND LIMIT SOLUTIONS solitons. We will study Ricci solitons in detail in a chapter of the successor to this volume. ...

GENERALIZED FIXED POINTS 23 EXAMPLE 2.3. Let (JRn, gcan) denote Euclidean space with its standard metric. Since the metric is ...

24 2. SPECIAL AND LIMIT SOLUTIONS Eternal solutions An eternal solution of the Ricci fl.ow is one that exists for all time. Su ...

ETERNAL SOLUTIONS 25 induced by the metric. One can also see the cigar's asymptotic approach to the cylinder in another way: t ...

26 2. SPECIAL AND LIMIT SOLUTIONS In particular, R"B = 0 ( e-^2 s) as s ---7 oo, which illustrates the exponential decay claimed ...

ETERNAL SOLUTIONS 27 So the Gauss curvature K is given by 2 r.p" ( s) (2.10) K = D 1 (e2, e1) = - <.p (s). Recalling equati ...

28 2. SPECIAL AND LIMIT SOLUTIONS Hence by (2.8), g must have the form 1 g = ds^2 + 2 tanh^2 (as) d(J2 a for some a > 0. We c ...