- EQUATIONS AND INEQUALITIES SATISFIED BY L AND £ 323
LEMMA 7.43 (L-distance and trace Harnack quadratic). Let f E (0, T).
Suppose"( : [O, f] ---+ M is a minimal £-geodesic from p to q. Then(7.78) 73 /^2 (R(q,f) + IX(f)l^2 ) = -K("!,f) + ~L(q,f).
Using (7.78), we can rewrite (7.57) and (7.55) as follows. For convenience
we include (7.76) below as (7.81).LEMMA 7.44. Let f E (0, T), let"(: [O, f] ---+ M be a minimal £-geodesic
from p to q, and let K = K ("!, f) be given by (7.75). Then, at (q, f),
oL 1 1
(7.79) -=-K--L+2 OT f 2f VT frR '2 - 4 2
IVLI =-4TR--K+-LVT VT'
(7.80)(7.81) b..L<--K+--2^1 n VT fR.
- r VT
These may seem like strange ways to rewrite equations (7.57) and (7.55).
A motivation is given by (7.76), which involves the integral K of Hamilton's
trace Harnack quadratic, which naturally arises from the second variation
formula for £-length. Note that besides K, these formulas do not explicitly
contain the quantity X. When the minimal £-geodesic is not unique, thequantities ~~, IV Ll^2 , b..L and Kare defined using the choice of "I·
6.2. Inequalities for L. Combining (7.79) and (7.80) with (7.81), we
getLEMMA 7.45. At (q,f) the L-distance function L (x,T) satisfies
oL 1 n
(7.82) - < -b..L - -L + -OT - 2f VT'
(7.83) oL - t::,.L + -
1- IV Ll^2 -
1
L - 2VTR + !!__ 2:: 0.
OT 2VT 2T VT
Equation (7.82) already exhibits the advantage of the L-distance over
the usual distance function in Riemannian geometry in the setting of the
Ricci fl.ow. It is a subsolution to a backward heat equation.
Recall that L (x, T) = 2./TL (x, T). Then at (q, f)
oL - < 2VT ( -b..L--L+- 1 n ) +-L= (^1) -t::,.L+2n -
OT - 2f VT VT '
so that
LEMMA 7.46 (L - 2nT supersolution of heat equation). At (q, f) the
function L (x, T) satisfies
(7.84)oL -
OT+ l::,.L:::; 2n,