1547671870-The_Ricci_Flow__Chow

EVOLUTION OF THE CURVATURE 109 D This formula simplifies nicely in the special case of a conformal defor- mation. COROLLARY 5. ...

110 5. THE RICCI FLOW ON SURFACES PROOF. Leth be a fixed metric on M^2 and let g be conformally related to h by g = euh. By Lemm ...

CURVATURE ESTIMATES USING RICCI SOLITONS 111 On the other hand, the ODE behaves much better when s 0 < min {r, O}, in which ...

112 5. THE RICCI FLOW ON SURFACES solitons. In particular, we shall obtain upper bounds for the scalar curvature by estimating s ...

CURVATURE ESTIMATES USING RICCI SOLITONS 113 obtain .. 1. (div M)i ~ V'^1 Mji = Y'jV'i\J^1 f - 2,V'iV'jV'^1 f k 1 1 =Rik V' f ...

114 5. THE RICCI FLOW ON SURFACES COROLLARY 5.14. Under the normalized Ricci flow on a compact sur- face, there exists a constan ...

CURVATURE ESTIMATES USING RICCI SOLITONS 115 To compute the evolution equation for l\7 fl^2 , we use equation (5.10) and recal ...

116 5. THE RICCI FLOW ON SURFACES for a short time. We call these Bernstein-Banda-Shi (BBS) derivative estimates. They originate ...

UNIQUENESS OF RICCI SOLITONS and its average scalar curvature by --'- fMn Rdμ P--c- V. If g is a Ricci soliton, then (2.3) imp ...

118 5. THE RICCI FLOW ON SURFACES Since R (x, t) :::; p (t) :::; 0, this is possible only if IRc-~gl2 = 0 at ( x, t). Tracing th ...

UNIQUENESS OF RICCI SOLITONS 119 where K = R/2 denotes the Gauss curvature of g. We now derive the form of (5.17) that we used ...

120 5. THE RICCI FLOW ON SURFACES Convergence when x (M^2 ) < 0 In this section, we will prove the following case of Theore ...

CONVERGENCE WHEN x (M^2 ) < 0 So for all t > 0 large enough, one gets :t IV' Rl 2 :S 6. IY' Rl 2 +~IV R l 2 ) whence the ...

122 5. THE RICCI FLOW ON SURFACES Define cp 2 J\7\7 RJ^2 by Then there exists ti 2 to large enough such that for all t 2 ti, one ...

CONVERGENCE WHEN x (M^2 ) = O 123 we obtain a lk/2J at (vkR) = 6. \7kR + L (vj R) ®g ( vk-j R) -r (vk R) j=O and hence ~ at l\ ...

124 5. THE RICCI FLOW ON SURFACES I I I I I I I I I ,,,t-I ------- I I I I I I I I I I I ---,~ ./ FIGURE 2. A surface of Euler ...

CONVERGENCE WHEN x (M^2 ) = 0 125 we obtain the inequality :t (t IV fl 2 + f 2 ) :S Ll (t IV fl 2 + f^2 ). Hence there is C1 = ...

126 5. THE RICCI FLOW ON SURFACES By the maximum principle, R + 2 IV f l^2 :S C /t for all positive times. Be- cause Proposition ...

CONVERGENCE WHEN x (M^2 ) = (^0 127) LEMMA 5.32. Let (M^2 , g (t)) be a solution of the Ricci flow on a closed surface with r ...

128 5. THE RICCI FLOW ON SURFACES Let N be a constant to be determined, and set <J? ~ tk+3 lvkRl2 + Ntk+2 lvk-l Rl2 Then ther ...