1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

242 33. NONCOMPACT HYPERBOLIC LIMITS

Jo (D^2 (1 + ry)). Since V (to) is normal to TA (to), we have that

(33.49) JV^0 (to)I = I~~ (to)ILoopo(to) I -+ 0 as to --+ oo.

Let x E int (V (to)). By definition (33.47) we have that x E int (Disk^0 (t)) for

t near to. If { e 1 , e 2 } is a basis for Tx Disk^0 ( t), then the induced area element on

Disk^0 (t) is

dA(t) = Jdet(J(t))w^1 /\w^2 ,

where

J (t) == ( g (t) (e1, ei) g (t) (e1, e2) )

· g(t)(e2,e1) g(t)(e2,e2)

and where {wi}~=l is the dual basis of 1-forms; i.e., wJ (ei) =of. Hence

at^8 dA (t) = 2 1 trace J(t) (f)J 8t (t) ) dA (t).

Choosing { e 1 , e2} to be orthonormal at time to, we obtain

:t lt=to dA (t) = ~ ( ( ~~ (to)) (e1, ei) + ( ~~ (to)) (e^2 , e^2 )) dA (to).

Hence, under the NRF the evolution of the area is given by

(33.50)

d

d I Areag(t) (Disk^0 (t)) = r ( ~ dA) (to)+ r v^0 (to)ds (to)

t t=to fv(to) ut JLoop^0 (to)

= ( (~r-Rc(e 1 ,e1)-Rc(e2,e2))dA(to)

fv(to)

+ r v^0 (to)ds (to),

JLoop^0 (to)

where ds (to) is the induced arc length element with respect to g(t 0 ).

Similarly to an argument in Schoen and Yau [354], we may bound the RHS. In

particular, at time to we have

(33.51) Re (e1, ei) +Re (e2, e2) =sect (e1 /\ e2) +sect (e 1 /\ e3)

+sect (e2 /\ ei) +sect (e2 /\ e3)

1

=

2

R +sect (e1 /\ e2),

where sect (e 1 /\ e2) denotes the sectional curvature of the plane spanned by e 1 and

Let II denote the second fundamental form. It follows from the Gauss equations

that the components of the Riemann curvature tensor of the disk V (to) are given

by

R~ke = R-t;1'ke + IIi£ IIjk -IIik IIje.

Thus the Gauss curvature K of V satisfies

(33.52) K = Rf 221 = R{-i 21 + IIn II22 - (II12)^2

= sect ( ei /\ e2) + det (II).