- LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 171
3. Local entropies are nontrivial near bad points
With the aim of obtaining a contradiction, we study the local entropies
centered at the bad points. In this section we shall prove the following.
STEP 3. A uniform negative integral upper bound for Vi over a ball.3.1. Local geometry at the bad points via adjoint heat kernels
and their local entropies.
To understand the (local) geometry of the solutions (Mf,gi (t)) based
at the well-chosen points (xi, ti) given by (21.21), we consider the (globally
defined) adjoint heat kernels centered at (xi, ti) and their corresponding
entropy functionals.
3.1.1. The adjoint heat kernels centered at (xi, ti).
Recall that the adjoint heat kernel centered at (xi, ti)
Hi : Mi x [O, ti) ---+ lR+
is defined to be the minimal positive fundamental solution of the adjoint
heat equation
(21.28a)(21.28b)
o; Hi ~ ( -! -b..gi + Rgi) Hi = 0,
liJ:Q. Hi(·, t) = 8xi
t/ti(see Definition 16.43 in Part II). Since the solution (Mi, gi (t)) is complete
with bounded curvature, such a solution to (21.28) exists; see Chapter 23.
By (26.20) in Chapter 26, we have
(21.29) { Hi (x, t) dμgi(t) (x) = 1
}Mi
for all t E [O, ti). Define fi : Mi x [O, ti) ---+JR by
(21.30) Hi (x, t) ~ (47r (ti - t))-n/^2 e-fi(x,t).
Define Perelman's differential Harnack quantity Vi on Mix [O, ti) by
(21.31) Vi ~ VHi = [(ti -t) ( Rgi + 21:::..gJi - IV' gJil^2 ) + fi - n J Hi.
By Perelman's differential Harnack estimate for complete solutions of the
Ricci fl.ow with bounded curvatures (for closed manifolds, see §9 of [152] or
Theorem 16.44 in Part II; for noncompact manifolds, see Theorem 7.1 and
Corollary 7.1 in Chau, Tam, and Yu [26]), we have
(21.32) Vi (x, t) ::::; 0 in Mi x [O, ti)·
Thus the entropies of the adjoint heat kernels are nonpositive (compare with
(17.44)):