1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. NOTES AND COMMENTARY 211

The claim now ensues from the following consequences of the co-area for-

mula:

This also completes the proof of the theorem. D

EXERCISE 22.18. Prove (22.98).

The following result has been used in deriving the contradiction in the

proof of the pseudolocality Theorem 21.9.

THEOREM 22.19. Let (Mn, g) be a Riemannian manifold and suppose

B (xo, p) is compact. If there exists a constant In E (0, oo) such that

(22.101) (Area(an)t:::::: In (Vol(n)r-^1

for any compact domain n C B (xo, p) whose boundary is 01 , then for any

01 function 'ljJ compactly supported in B (xo, p) we have

JM (2jV''lj;j2 - 'lj;2 log'l/J2) dμ +log (JM 'l/J2dμ) JM 'l/J2 dμ

(22.102) :::::: (sn +log (:))JM 'lj;^2 dμ.

PROOF. By approximation, we may assume that 'ljJ is a nonnegative 01

function compactly supported in B (xo, p). With this, the proof of Theorem

22.16 applies without change. Note that since supp ('lj;) c B (x 0 , p), we have

Ms C B(xo,p) for s > 0. D

5. Notes and commentary

§4. For a classical application of spherical symmetrization to the proof of
the Faber-Krahn inequality (originally conjectured by Rayleigh), see §III.3
of Chavel [28].

Backward uniqueness and unique continuation. Recently, related to work
of Alexakis [2] (see also Alexakis-Ionescu-Klainerman [3] and Wong-Yu
[190]) and using Carleman-type estimates, Kotschwar [112] proved the fol-
lowing.

THEOREM 22.20 (Backward uniqueness for solutions of Ricci fl.ow). If

gl (t) and g2 (t) are two complete solutions of the Ricci flow with bounded
curvature on a manifold Mn and time interval [O, T] such that gl (T) =
g2 (T), then gl (t) = g2 (t) for all t E [O, T].

As a consequence, we have the following answer to a question of Arthur
Fischer (see also Problem 4.21 in [45]).