- THE MEAN CURVATURE FLOW 485
Tracing the above formula, we see that the mean curvature evolves byaH a .. a
at = -at9ij · hij + lJ at hij
= 2H lhijl^2 + ( t:::.H - H jhijl^2 + H (Rep )vv)
= t:::.H + H jhijl^2 + H (Rcp)w.
This is (B.17).
Equation (B.18) follows from (B.13) anda 1 .. (a )
at dμ = 2lJ at9ij dμ.Finally we rewrite the evolution equation for hij to see (B.16). Recall
that the Gauss equations say that for any tangent vectors X, Y, Z, Won M,
(RmM) (X, Y, Z, W) = (Rmp) (X, Y, Z, W)
+ h (X, W) h (Y, Z) - h (X, Z) h (Y, W).Tracing implies (in index notation)(RcM)ie = (Rcp)u 1 - (Rmp)viev + Hhie -h7e·
The Codazzi equations say for any X, Y, Z ET M,
(\Jxh) (Y, Z) - (\Jyh) (X, Z) = - (Rmp (X, Y) Z, v).In index notation, this is
\Jihkj = \Jkhij - (Rmp)ikjv ·
Tracing over the Y and Z components in the hypersurface directions, we
have
· \JH-div(h)=-Rcp(v).To obtain (B.16) from (B.15) we note that, by using the Codazzi equations
and Gauss equations, we obtain(B.20)where Rep (v) ~ (Rcp)jvdxj is a 1-form on M. Note that
v 1 ;ax ax)
(rp )ij = -21/ \ axi' axj = -hij
and similarly