1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE MEAN CURVATURE FLOW 487

EXERCISE B.9. Compute gt (lhJ^2 - ~H^2 ). See §5 of Huisken [212] for

a study of under what conditions on (P, gp) the pinching estimate

JhJ2 - !._H2:::; C. H2-o

n

holds for some 8 > 0 and C < oo.

EXERCISE B.10. Let Xt : Mn ---+ pn+l be a hypersurface evolving by

mean curvature fl.ow where the metric gp (t) on P evolves by Ricci fl.ow.

Compute gt hij.

HINT. The terms -Di (Rep )jv -Dj (Rep )iv +Dv (Rep )ij are cancelled
by new terms introduced by the Ricci fl.ow. Note that the above terms
represent the evolution of the Christoffel symbols under the Ricci fl.ow.

2.2. Huisken's monotonicity formula for mean curvature flow.
When the ambient manifold pn+l is euclidean space JRn+l, Lemma B. 7 says
the following.

LEMMA B.11. The evolution of the first fundamental form (i.e., induced

metric), normal, and second fundamental form are given by the following:

[)

(B.21) atgij = -2Hhij,

(B.22)

(B.23)

(B.24)

(B.25)

Daxll at = \JH,

[) 2

at hij = !:l.hij - 2Hhikhkj + JhJ hij,

!H = !:l.H + JhJ

2

H,

[) 2

atdμ = -H dμ.

For the mean curvature fl.ow there is a monotonicity formula due to
Gerhard Huisken (see [213], Theorem 3.1).

THEOREM B.12 (Huisken's monotonicity formula). Let Xt : Mn ---+

ffi.n+l, t E [O, T), be a smooth solution to the mean curvature flow (B.11).
Then any closed hypersurface Mt~ Xt (M) evolving by MCF satisfies
(B.26)

.:!:__ { (4?Tr)-~e-^1 ~;

2

dμ=-{ (H- (X,v))

2

(47Tr)-~e-qt_dμ,

dt}Mt }Mt 2r

where r ~ T - t.

n IX1^2

PROOF. Let u ~ (47Tr)-2 e-~. Using (B.11) and (B.25), we compute

(B.27) .:!:_ { udμ = { (!! + H (X, v) - JXJ2 - H2) udμ.

dt}Mt }Mt 2r 2r 4r^2