1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

FIRST PROPERTIES OF THE BACKWARD LIMIT v 00 81 Since J 1 (g(t)) E [-C,O) fort E (-oo,-1] and since -d d J1 (g (t)) =^1 ( -2R^2 ...

82 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Recall that from (29.7) we have (29.49) Hence we have that (29.50) DEFINITION 29. ...

ISOPERIMETRIC CONSTANT OF METRICS ON S^2 83 Isoperimetric constant of metrics on 52 An important result for the Ricci flow on 5 ...

84 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS PROOF. (1) This follows easily from considering small distance circles; see Lemma ...

ISOPERIMETRIC CONSTANT OF METRICS ON S^2 85 The curve shortening flow is the case where f N = -r;,N is the curvature vector of ...

86 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS We thus obtain from (29.59) and Li(ti):::; C that J,i t, ( d~7f) 2 ds:::; c. From ...

CHARACTERIZING ROUND SOLUTIONS 87 where r = d(x,p). Using this, we compute that L(r) = 27rc - iR(p)c^3 + O(c^4 ), Area(Bp(c)) ...

88 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Since the function t HI (t) may not be differentiable, given a time f, we define ...

CHARACTERIZING ROUND SOLUTIONS 89 each 'Yt, is a minimizer for I ( ·; g(ti)). Consider the pointed sequence of solutions (5^2 ...

90 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS A consequence of the above proposition is the following. LEMMA 29.23 (Areas of th ...

CHARACTERIZING ROUND SOLUTIONS 91 to (N 00 ,9 00 (0),p 00 ) that Oi converges to a point 000 E loo C N 00 and that /3Hsi) conv ...

92 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS We first prove the following result. LEMMA 29.25 (Connectedness of distance circl ...

CHARACTERIZING ROUND SOLUTIONS c:-neck, we have f3 n C 0 = 0. Let u E (3. Since f3 c 5; o, +' we have d(u,a)::::; d(u,p) + d(p ...

94 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS assume that s(<p(f3z(t)));:::: s for all t E [1, £ 1 ). Define L2 to be the fi ...

8. CHARACTERIZING ROUND SOLUTIONS 95 for 0 :::; r1 :::; r2 :::; r3 :::; r4. Hence, for r 2 2, Areag(tJ(B gCtl(r-1)) 2 (r ~rl) 2 ...

96 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS LEMMA 29.29. There exists a constant c 1 > 0 such that for all t E (-oo, -1], ...

CHARACTERIZING ROUND SOLUTIONS 97 8.4. Proof of the main proposition on characterizing round solutions. First, we observe a ba ...

98 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS that for any c > 0 there exists Tc :::::; - 1 such that g(t) 2". C^1 g 52 fort ...

CHARACTERIZING ROUND SOLUTIONS 99 Indeed, suppose that rt C Bg(t) (k) for some t E (-oo, t~]. By (29.94) we have Thus A+bt) &l ...

ioo 29. C OMPACT^2 -DIMENSIONAL ANCIENT SOLUTIONS '' ' ' '' ' ' FIGURE 29.5. 5g_(t)(s x, t ) (^59) x, (t)(r t ) By (29.97), ther ...