1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

482 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

2. The mean curvature fl.ow

In this section we give a brief introduction to the mean curvature flow

and some monotonicity formulas. It is interesting to compare these mono-

tonicity formulas for the mean curvature flow to those for the Ricci flow.

We also refer the reader to the books by Ecker [132] and X.-P. Zhu [386].

2.1. Mean curvature fl.ow of hypersurfaces in Riemannian man-

ifolds. Let (pn+l, gp) be an orientable Riemannian manifold and let Mn

be an orientable differentiable manifold. The first fundamental form of an

embedded hypersurface X : M --> P is defined by

g (V, W) ~ gp (V, W)

for V, W E TxX (M). More generally, for an immersed hypersurface, we

define

g(V, W) ~ (X*V,X*W)

for V, W E TpM. Let v denote the choice of a smooth unit normal vector

field to M. The second fundamental form is defined by

h (V, W) ~ (Dvv, W) = -(DvW, v)

for V, WE TxX (M), where D denotes the Riemannian covariant derivative

of (P, gp) and ( , ) ~ gp ( , ). To get the second equality in the line above,

we extend W to a tangent vector field in a neighborhood of x and use

(v, W) = 0. In particular, 0 = V (v, W) = (Dvv, W) + (DvW, v). The

mean curvature is the trace of the second fundamental form:

n

H~ Lh(ei,ei),

i=l

where { ei} is an orthonormal frame on X (M).

A time-dependent immersion Xt = X (·, t) : M --> P, t E [O, T), is

a solution of the mean curvature fl.ow (MCF) of a hypersurface in a

Riemannian manifold if

ax ~

(B.11) fit (p, t) = H (x (p, t)) ~ -H (p, t) · v (p, t), p EM, t E [O, T),

where ff is called the mean curvature vector. When the Xt are embed-

dings, we define Mt ~ Xt (M). From now on we shall assume that the Xt

are embeddings, although for the most part, the following discussion holds
for immersed hypersurfaces.

Let {xi} ~=l denote local coordinates on M so that { ~-;; } are local

coordinates on Mt. We have

9ij ~ g ( ~~' ~~) = ( ~~' ~~) '

hij ~ h (

8

8

~,

8

8

~) = jaa~,Daxv) =-/Dax

8

8

~,v).

Xi xJ \ Xi DxJ \ axi xJ