1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

HOMOGENEOUS EXPANDING SOLITONS 37 Define a vector field 1 1 1 X = -2x1F1 - 2x2F2 - (2x1x2 + x3)F3. A calculation shows that th ...

38 1. RICCI SOLITONS The Ricci tensor of this metric is simply Rc(g) = -8(w^1 0 w^1 ). Given anyμ E JR, define a vector field X ...

HOMOGENEOUS EXPANDING SOLITONS 39 many details. To discuss convergence of Ricci fl.ow solutions, it is necessary to anticipate ...

40 1. RICCI SOLITONS With respect to the frame a(t), one obtains the identification ( ~r4/3 o o ) 9a(t) = 0 1r2/3 0. 0 0 lt-2/3 ...

7. When breathers and solitons are Einstein Using the dual field w^1 = dx1, w^2 = dx^2 - x1dx^4 , w^3 = dx3 - x2dx^4 , define a ...

42 1. RICCI SOLITONS 0 with equality only if Re = 0. Thus, if Rmin < 0, then R is constant at this 0 time. By applying the sa ...

WHEN BREATHERS AND SOLITONS ARE EINSTEIN 43 REMARK 1.69. A version of Corollary 1.68 for the Ricci tensor may be proved direct ...

44 1. RICCI SOLITONS Because of the fl.at factor, R^2 = 2JRcJ^2 , and therefore dr/dt = r^2 /3. This implies that r increases wi ...

PERELMAN'S ENERGY AND ENTROPY IN RELATION TO RICCI SOLITONS 45 Chapter 6 and the ·associated equations for g, f and 7 > 0. ...

46 1. RICCI SOLITONS THEOREM 1.75 (Expander entropy monotonicity formula). For a solu- tion (g (t), f+ (t), r(t)), r (t) > 0, ...

BUSCHER DUALITY TRANSFORMATION 47 and {ya:n=l be local coordinates on Mand P, respectively, such that ha/3 ~ h ( aia, a~f3) = ...

48 1. RICCI SOLITONS LEMMA 1.79. The scalar curvature o{t>g and ttg are given by b R = R-q.6.logA-q (q: l) IVlogAl^2 , ttR=R+ ...

BUSCHER DUALITY TRANSFORMATION 49 satisfy (1.82) where U¢~ b¢+f. REMARK 1.82. Note that b\7 af = tt\7 af = 0. The functions b ...

50 1. RICCI SOLITONS 9.3. Examples. The Buscher dual of a rotationally symmetric soliton solution on a surface I> g = dr^2 + ...

SUMMARY OF RESULTS AND OPEN PROBLEMS ON RICCI SOLITONS 51 10.1. Gradient Ricci solitons on surfaces. The following is a com- p ...

52 1. RICCI SOLITONS PROBLEM 1.88. Are there any 3-dimensional steady gradient solitons besides a flat solution, the Bryant soli ...

NOTES AND COMMENTARY 53 Section 4. Einstein metrics are special cases of Ricci solitons. In gen- eral, if a warped product met ...

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Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons Symmetry, as wide or narrow as you may define its meaning, is one idea by ...

56 2. KAHLER-RICCI FLOW are holomorphic (complex analytic), and hence biholomorphic, whenever ui n Uj i= 0. Two systems of holom ...