1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. EQUATIONS AND INEQUALITIES SATISFIED BY L AND e 327

LEMMA 7.50 (Reduced distance - partial differential inequalities). At

(q, f) the reduced distance .e (x, T) satisfies

(7.91)

a.e 2 n

87

- ~.e + 1v.e1 -R +

27

2:: o,

(7.92)

2 £-n

2~£ - IV£1 + R + -_-:::;; 0,

T

a.e .e n

(7.93) -+~.e+---<O 8T f 27-'

(7.94)

a.e 2 .e

287 +1\7£1 -R+:y:=O,

lim £(q,f) =1

7'-->0+ [ dg(O) (p, q) J 2 I 4f '

(7.95) min£(q,f) =

4

1

_ minL(q,f):::;; ".!'..

qEM T qEM 2

REMARK 7.51. (i) Note that (7.94) is the only possible equality obtain-
able from (7.88) and (7.89) which does not involve K.
(ii) In the inequalities above, the direction of the inequality depends on
the sign of the coefficient in front of the term ~.e. The reason for this is that,
analogous to considering ~ ( d^2 ) on a Riemannian manifold with nonnegative
Ricci curvature, ~£has an upper bound (7.90).

The above equations for .e demonstrate, in the context of Ricci fl.ow, the
superiority of the reduced distance over the Riemannian distance function.
It is a space-time notion of distance which is a subsolution of Laplace-type
and heat-type equations. Note that (7.91) is a forward heat superequation
whereas (7.93) is a backward heat subequation. One would think that the
backward heat subequation is more natural since it is associated to the
backward Ricci fl.ow, but we shall find the forward superequation (7.91)
very useful.

REMARK 7.52. (i) If we replace the inequality by an equality in (7.91),

we obtain equation (6.14) for fused in the study of the entropy W:

a I 2 n

8T f - Lf + \7 f I - R + 2T = 0.

(ii) Compare (7.92) with the equation (6.52):

2 L f - IV !1^2 + R + f - n = !μ (g, T)

T T

for the minimizer f of W (g, ·, T).

If we divide by 2./T in the LHS of (7.65), we get (\7\7£ +Re) (Y, Y) at
smooth points of £. Motivated by the special case of shrinking solitons, we