198 5. ENERGY, MONOTONICITY, AND BREATHERS
2.1. A coupled system equivalent to the gradient flow of ;:m.
There is a coupled system, i.e., (5.29)-(5.30), induced from the gradient
fl.ow (5.25)-(5.26) obtained simply by computing the evolution equation for
f =log (dμ/dm). As we shall see, this coupled system is equivalent to the
gradient fl.ow.
LEMMA 5.12 (Measure-preserving evolution off under modified RF).
The function f(t) in a solution (g (t), f (t)) of the gradient flow of P (5.25)
and (5.26) satisfies the following equation:
af
at = -b.f-R.DRelated to the above calculation, we have the following.
EXERCISE 5.13. Show that if w1 (t) and w2 (t) are time-dependent n-
forms, then
!!log (w1) = /Jtwl /Jtw2,
at W2 W1 W2
where the quotient of two n-forms is defined as in Remark 5.9.
Hence we consider the coupled modified Ricci flow
a
(5.29) at9ij = -2(Rij + \7i\7jf),
af
(5.30) at = -b.f - R.Note that the first equation is a modified Ricci fl.ow equation whereas the
second equation is a backward heat equation.
LEMMA 5.14. The coupled modified Ricci flow equations (5.29)-(5.30)
are equivalent to the gradient flow (5.27).
PROOF. If g (t) is a solution to (5.27), then by Lemma 5.12, (g (t), f (t)),where f =log (dμ/dm), is a solution to the system (5.29)-(5.30).
Conversely, if (g (t), f (t)) is a solution to the system (5.29)-(5.30), then
dm ~ e-f dμ satisfies
:t(dm)= (-~~ -R-b.f)e-fdμ=O;
that is, g (t) is a solution to (5.27) with dm as defined above. D