288 7. THE REDUCED DISTANCE
EXERCISE 7.2 (£-distance and .e between two space-time points). Show
using (7.1) that inf 7 £ 71 , 72 ('Y), where the infimum is taken over all C^1 -paths
/ with /(T1) = p and 1(T2) = q, is attained by/ (T) = /3 (2y'T), where
f3 : [2y17i, 2..fTi] ~ M is a minimal geodesic with constant speed I~~ I§ =
d(y,q) H
2 y172_ 2 VT1. ence( ) 7.8 L (p 71 ) ( q, T2 ) =:= .. m f 0Ti,T2 C' ( I ) = d (p, q)2
' 'Y^2 yT2 r;:;::;; -^2 yTl r;::::We may then define the reduced distance by
( )
.e ( ). L(p,T1) (q, T2) d (p, q)
2(^7) · (^9) (p,Ti) q,T (^2) =;= 2y1Ti+2yl7i= 4(T2-T1)'
We compute under the assumption Rc_g ;::: 0
(7.10) (% 7 + ~) (L-2n7) = ~ (d^2 ) - 2n = 2 ( d~d+ l\7dl
2
-n) :S 0
in the weak sense,^3 where we used l\7dl = 1 a.e. and the Laplacian Com-
parison Theorem (A.9): d~d :Sn - 1. That is, L - 2n1' is a subsolution of
the backward heat equation in the weak sense.REMARK 7.3. It is useful to keep in mind the examples of flat tori to
see why one cannot prove a stronger statement (see subsection 9.5 of this
chapter).
The role of the reduced distance in the study of the heat equation is
exhibited by the Li-Yau differential Harnack inequality (A.12), which im-
plies that for any positive solution u of the heat equation on a complete
Riemannian manifold with Rc_g;::: 0,
u(x2, T2) (T2 )-n/
2( ) 2: - exp { -.e(x }
U Xl, Tl Tl^1 '^71 ) (x2, T2) ,where T1 < T2. A similar, but slightly more complicated, statement holds
when Rc_g;::: -Kg, where K;::: 0. See [253] for details.
2. The £-length and the £-distance
Let (Nn,h(t)), t E (a,w), be a solution to the Ricci flow. From this
we can easily obtain a solution (Nn, h (T)) to the backward Ricci fl.ow
8
07h = 2Rc
(^3) That is, for any nonnegative 02 function rp on space-time with compact support,
1
00
JM (-~~ + 6.rp) (L-2nf) dμdf ~ 0.
See subsection 9.5 of this chapter for a justification of (7.10).