1547671870-The_Ricci_Flow__Chow

AN INJECTIVITY RADIUS BOUND 149 for T E [t - 1, t]. Let '"'( be a geodesic joining x and x 1 of constant speed with respect to ...

150 5. THE RICCI FLOW ON SURFACES PROPOSITION 5.65. Let (M^2 ,g(t)) be a solution of the normalized Ricci flow on a closed surfa ...

AN INJECTIVITY RADIUS BOUND 151 PROOF. Choose d < njJKmax so that max{L(,Bi)} < d. Let f be the space of all nondegenera ...

152 5. THE RICCI FLOW ON SURFACES length less than 7r/VKmax· Then since L(a1) = L(a2) < 7r/VKmax, there are by Lemma 5.66 uni ...

THE CASE THAT R (-, 0) CHANGES SIGN 153 COROLLARY 5.72. Let "ft be a smooth I -parameter family of loops in a solution (M^2 ,g ...

154 5. THE RICCI FLOW ON SURFACES injectivity radius estimate of Proposition 5.65, we proceed as in Proposi- tion 5.51 to obtain ...

13. THE CASE THAT R(·,O) CHANGES SIGN 155 Since Area (M^2 , g (t1)) = Area (M^2 , go), this implies the desired uniform upper bo ...

156 5. THE RICCI FLOW ON SURFACES As in Proposition .5.62, the desired positive lower bound for R - s will follow from (5.46) on ...

MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 157 REMARK 5. 78. The result that a sequence of blow-ups (M^2 , 9J ( t)) pulled bac ...

158 5. THE RICCI FLOW ON SURFACES have Cs (r) =Lg (r) 2 ( Ag (Mi)^1 +Ag (M§)^1 ) (5.47) 2 Ag (M2) =Lg (r) Ag (Mi) ·Ag (M§)' wher ...

MONOTONICITY OF THE ISOPERIMETRIC CONSTANT i59 For E > 0 small enough, '"Ye is a smooth embedded loop. Since g is Euclidean ...

160 5. THE RICCI FLOW ON SURFACES SKETCH OF PROOF OF LEMMA 5.85. Let L and A denote length and area, respectively, measured with ...

MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 161 area 'outside' of 'Yoo· We then have CH ('Yoo)= (L1 + L2) 2 (~ 3 + A 1 : A 2 ) ...

162 5. THE RICCI FLOW ON SURFACES We are now ready to state the main assertion of this section. Notice that it does not matter w ...

14. MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 163 where K = R/2 is the Gaussian curvature and k = (T, 'VrN) is the geodesic cur ...

164 5. THE RICCI FLOW ON SURFACES LEMMA 5.92. If (M^2 , g (t)) is a solution of the Ricci flow on a topological 2-sphere, the is ...

AN ALTERNATIVE STRATEGY FOR THE CASE x (M^2 > 0) 165 PROOF OF THEOREM 5.88. We shall show that for any E > 0, we have :t ...

166 5. THE RICCI FLOW ON SURFACES For the remainder of this section, we shall adopt notation different from that used in the res ...

AN ALTERNATIVE STRATEGY FOR THE CASE x (M^2 > 0) 167 denote the area of M;. Since 'Y is a geodesic, the Gauss-Bonnet formul ...

168 5. THE RICCI FLOW ON SURFACES IR (xi, ti) I maxM2 IR(., ti) I and consider the sequence of parabolically dilated solutions ( ...