1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE MEAN CURVATURE FLOW 483

Note that H = gij hij and

(B.12)

We have the following basic formulas for solutions of the mean curvature

fl.ow.

LEMMA B. 7 (Huisken). The evolution of the first fundamental form (in-

duced metric), normal, and second fundamental form are given by the fol-

lowing:

(B.13)

(B.14)

(B.15)

(B.16)

(B.17)

(B.18)

a

otgij = -2Hhij,

D .E_V = \l H,

8t

ot a hij = \l i \l j H - Hhjkg ke hei + H (Rm p )z;ijz;

= D..hij - 2Hhikhkj + ihi^2 ~j

+ hij (Rcp)w - hik (Rcp)jk - hkj (Rcp)ik

+ 2 (Rmp hij£ hke + hki (Rmp )z;kjz; + hkj (Rmp )z;kiz;

• Di (Rep )jz; - Dj (Rep )iz; +Dz; (Rep )ij,

:tH = D..H + ihi^2 H + H (Rcp)w,

-dμ=-Hdμ a^2

8t '

where \l is the covariant derivative with respect to the induced metric on the

hypersurface and dμ is the volume form of the evolving hypersurface.

REMARK B.8. Technically, we should consider these tensors (or sections

of bundles) as existing on the domain manifold M. However, we shall often

view them as tensors on the evolving hypersurface Mt = Xt (M). The

time-derivative of the unit normal is expressed slightly differently since it
is actually the covariant derivative of v in the direction ~~ along the path

t r--t Xt (p) for p E M fixed. On the other hand, if we view g and h as on

the fixed manifold M, we have the ordinary time-derivatives, whereas if we

view g and h as on the evolving Mt, then the time-derivatives are actually

Da. Ft

PROOF. While carrying out the computations below, keep in mind that
the inner product of a tangential vector with a normal vector is zero. The