1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

EXAMPLES OF KAHLER-RICCI SOLITONS 97 For some other recent work on the Kahler-Ricci fl.ow the reader is re- ferred to Phong-St ...

98 2. KAHLER-RICCI FLOW are mutually exclusive: applying the Futaki functional F[w] to the holo- morphic vector field X = grad f ...

EXAMPLES OF KAHLER-RICCI SOLITONS 99 ]pm-l = (II~=l cpa(U:i<))/ ~,where, for example, cp1 ( U1 ) 3 ( z1, ... , Zn-1 ) ~ ( - ...

100 2. KAHLER-RICCI FLOW and 32 Ra73 = - f)zaf)zf3 log <let g, exactly as in (2.6). If Q denotes the soliton potential functi ...

EXAMPLES OF KAHLER-RICCI SOLITONS 101 whose solution is cpr = cp^1 -neμ,r.p(v +>..In+ nin-1), where v is another arbitrary ...

102 2. KAHLER-RICCI FLOW P(r) on en{O} for those whose behavior at the boundary izi = 0 implies that l_: the metric is completed ...

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 103 space of L7:_ 1 is simply en blown up at the origin.) As JzJ ---+ ...

104 2. KAHLER-RICCI FLOW PROPOSITION 2. 79 (Evolution equation for the curvature). Under the Kahler-Ricci flow, (2.100) ( :t -~) ...

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 105 This implies (2.103) where we used (2.102). Since (2.103) is tens ...

106 2. KAHLER-RICCI FLOW PROOF. Indeed, substituting fJ =a and J = 1 in (2.100), we have (! -.6.) Raa'Y"f = Rap,v:yRμa"fiJ - Rap ...

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 107 PROOF OF THEOREM 2. 78. (1) We first prove that the nonnegativity ...

108 2. KAHLER-RICCI FLOW where we used l::~=l Re (Rap,Rμaaii + Rap,Raaμa) = 0 at (xo, to). On the other hand, by (2.109) with a= ...

MATRIX DIFFERENTIAL HARNACK ESTIMATE 109 For the ·proof of this lemma, which is elementary in nature, we refer the re~der to M ...

no 2. KAHLER-RICCI FLOW When Re > 0, the (1, 0)-vector minimizing the LHS of (2.113) is X^7 = (Rc-^1 )^7 ,o.7,oR, where (Rc- ...

MATRIX DIFFERENTIAL HARNACK ESTIMATE 111 Before we prove the corollary, we first recall how to go from the Kahler- Ricci flow ...

112 2. KAHLER-RICCI FLOW by taking 'Y (t) to be a minimal geodesic, with respect to g (ti), joining x1 to x2 with speed l'Y(t)i ...

MATRIX DIFFERENTIAL HARNACK ESTIMATE 113 In particular, if x E Bt (y, 1) , then R (x, t) :::; n (1 + e-^1 ) exp ( 4 (l ~ e-l)) ...

114 2. KAHLER-RICCI FLOW 9.3. Proof of the matrix Harnack estimate. In this subsection we prove Theorem 2.87. Let The following ...

MATRIX DIFFERENTIAL HARNACK ESTIMATE 115 where Wa = 9f3a Wf3, then (2.119) implies ( :t -~) (zai3WaW,e) = (Rai3'YJZ^0 1) WaW,e ...

116 2. KAHLER-RICCI FLOW PROOF. Let ha/j be a real (1, 1)-tensor. Using (2.33), we compute :t (\7 i\17ha/j) = \7 ;s'\J 7 ( :t ha ...