1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

INTRODUCTION TO THE KAHLER-RICCI FLOW 77 Differentiating (2.3), we find that for both the Kahler-Ricci ft.ow and the normalize ...

78 2. KAHLER-RICCI FLOW REMARK 2.41 (1-dimensional normalized Kahler-Ricci flow). Recall the following facts, due to Hamilton [1 ...

INTRODUCTION TO THE KAHLER-RICCI FLOW 79 COROLLARY 2.43 (Estimate for f). If Mn is closed, then, for the func- tion f given by ...

80 2. KAHLER-RICCI FLOW Here is an interesting equation due to Hamilton. LEMMA 2.47 (Ricci soliton vanisher evolution equation). ...

EXISTENCE AND CONVERGENCE 81 5.1. Cao's existence and convergence theorem. When the first Chern class has a definite sign, eit ...

82 2. KAHLER-RICCI FLOW Moser iteration, and differential Harnack estimates (see the discussion be- low). The corresponding theo ...

EXISTENCE AND CONVERGENCE 83 If r =/= 0, the integral on the RHS is equal to C~n ( e;;t - 1). When r = 0, we obtain Cit. Simil ...

84 2. KAHLER-RICCI FLOW By the uniform bound (2.43) on f, we conclude for some C 2:: i. 1 det (ga13(x, t)) - -=< <C C - de ...

EXISTENCE AND CONVERGENCE 85 constant C' < oo. Thus a bound on Y shall imply an estimate of the complex Hessian of <p, i ...

86 2. KAHLER-RICCI FLOW (using the Kahler identities (2.2) to get the second equality in (2.67)). Note that We claim that the Ca ...

EXISTENCE AND CONVERGENCE 87 (1) (Uniform estimate when c1 ::::; 0) If c1 (M) ::::; 0, then there exists a constant C < oo ...

88 2. KAHLER-RICCI FLOW To get a bound for w independent of T when r ::; 0, we need to work harder. When r ::; 0, we shall use t ...

EXISTENCE AND CONVERGENCE 89 on the time interval [O, T). Once we have a uniform C^2 '°'-estimate of <p on [O, T) (we shall ...

90 2. KAHLER-RICCI FLOW simplicity we drop the indices k in our notation below. Since g is Kahler, we may write 9a/J locally as ...

EXISTENCE AND CONVERGENCE 91 books on PDE, such as Moser [276], Morrey [274], Gilbarg and Trudinger [155], Han and Lin [195] ( ...

92 2. KAHLER-RICCI FLOW LEMMA 2.63 (Linear algebra). There exist unitary vectors rl, ... , rN E en with the property that, as He ...

EXISTENCE AND CONVERGENCE 93 On the other hand, applying (2.85) to bound the LP-norm of Mv(2) - wv, we have for each v :2: 2, ...

94 2. KAHLER-RICCI FLOW Putting these together, we obtain 1 w1 (2R) ~ (Vol ~(R) k(R) (Mi (2) - m1 (2) )Pr 1 :<; (Vol ~(R) k(R ...

SURVEY OF SOME RESULTS FOR THE KAHLER-RICCI FLOW 95 parabolic Schauder estimate (e.g., Theorem 5 on p. 64 of Friedman [146]) t ...

96 2. KAHLER-RICCI FLOW THEOREM 2. 70 (KRF: compact positive bisectional curvature). Suppose (Mn, g ( 0)) is a closed Kahler man ...