1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. INCOMPRESSIBILITY OF BOUNDARY TORI 241

LoopO(t)

FIGURE 33.2. Disk^0 (t) and Loopo (t).

The bound for d+d~' (to) involves the normal velocity of Loop^0 (t 0 ), which we

first discuss. Let N (to) be a choice of smooth unit normal vector field to TA ( t 0 ),

with respect to g(to). Since the embedded torus TA(t) depends smoothly on t , for

E: > 0 small enough and for each t E (to - E:, t 0 + c) , TA (t) can be uniquely written

as an exponential normal graph over TA (to). That is, there exists a unique smooth

function</>: TA (to) x (to - E:, to+ c) ---+IR such that each map 4?t : T A (to) ---+TA (t)

defined by

4?t (x) = exp~(to) (</> (x , t) N (to))

is a diffeomorphism. We call ~f ( ·, t 0 ) the normal velocity function of TA (t 0 ).

Since MA (to) converges to (HA, hl"HJ, we have that inside MA (to) CM,

uniformly in x as t 0 ---+ oo for each k 2 O; here r is the average scalar curvature. So,
since the boundary component TA (to) is a CMC torus of fixed area A, we conclude
that the normal velo city function %f; of TA (to) in M tends to zero as t 0 ---+ oo.

Let v^0 (to) : Loop^0 (to) ---+IR denote the normal velocity function of Loop^0 (to)

with respect to the outward unit normal of the boundary of Disk^0 (to) c