1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE MEAN CURVATURE FLOW 489

Formula (B.26) is useful for studying Type I singularities of the mean

curvature fl.ow, where

sup (T - t) lhl^2 < oo.

Mnx[O,T)

For Type I singularities there is a natural way to rescale the MCF equation:

X ~( p, tJ f\ =:= y'^1 X (p, t), where t ~ (t) =:=log V7'1=t.^1

2 (T-t) T-t

Then M-= t. X-(Mn) t '

dX ~ ~

-~ =-Hv+X

dt

and (B.26) becomes the normalized monotonicity formula:

From this one can prove the following (see Proposition 3.4 and Theorem 3.5
in [213]).

THEOREM B.14 (MCF convergence to self-similar). Suppose Xt: Mn--+

JRn+l, t E [O, T), is a Type I singular solution to the mean curvature fiow.

For every sequence of times ti --+ oo, there exists a subsequence such that

Mt;i converges smoothly to a smooth immersed limit hypersurface M 00 which
is self-similar:

Huisken's monotonicity formula was generalized by Hamilton [184] as
follows. Let Xt : Mn --+ pN, t E [O, T), be a submanifold of a Riemann-
ian manifold (PN, g) evolving by the mean curvature fl.ow^1 gtx = ii and
suppose u : pN x [O, T) --+ IR is a positive solution of the backward heat
equation:

Then

&u

&t = -1:::.pu.

!!__ [(T - t)(N-n)/2 { udμl

dt Jxt(M)

= - (T-t)(N-n)/^2 r Iii -(\71ogu)Jf udμ

Jxt(M)

-(T-t)(N-n)/^2 { tr_l(\7\7logu+ l g)udμ,

Jxt(M) 2 (T-t)

where dμ is the volume form of the submanifold Xt (M) and tr J_ denotes

the trace restricted to the normal bundle of Xt (M) C pN. By Hamilton's

1 In all codimensions the mean curvature vector Ji is defined by tracing the second
fundamental form, which takes values in the normal bundle.