- INCOMPRESSIBILITY OF BOUNDARY TORI 243
Let ""1 and ""2 denote the eigenvalues of II, i.e., the principal curvatures of
V (to). Since V (to) is minimal, we have "" 1 + "" 2 = 0 so that det (II) = -J<i;i ::::; 0.
Hence , by the Gauss- Bonnet formula and since x(V (to))= 1,(33.53) ( sect(e 1 !\e2)dA(to):::'.'. ( KdA(to)=21f- ( kds(t 0 ) ,
Jv(to) Jv(to) Jav(to)
where k denotes the geodesic curvature of the curve 8V (to) in the surface V (t 0 ).
From (33.48), (33.50), (33.51), and (33.53), we conclude that
LEMMA 33.27. Under the hypotheses of Proposition 33.24 and for to sufficiently
large, we have(33.54) d+dAJ (to)::::; -21f+ ( (k + v^0 ) ds (to)+ ( (~r -~R) dA (to),
t J 8V(t 0 ) Jv(to) 3 2
where the area of V (to) E J (to) equals A J (to).
We want to estimate the last two terms on the RHS of (33.54). It follows from(32.22), which says limt-+oo Rmin (t) = limt-+oo r (t) = -6, that given any E > 0,
there exists t" ( 1) < oo such that for all t 0 :::'.'. t" ( 1),
(33.55) ( (~r -~R) dA (to)::::; ( (~r -~Rmin) dA (to)
Jv(to) 3 2 Jv(to) 3 2
::::; - (1 - c) Areag(to) (V (to))= - ( 1 -E) A J (to).
To understand the term fav(to) kds (to) in (33.54), we first co nsider the modelcase of an exact hyperbolic cusp ([O, oo) x V^2 , dr^2 + e-^2 r 9flat), where (V, 9flat) is a
flat 2-torus. Suppose that we have a loop L which lies inside a slice {r} x V andwhich bounds a surface E^2 c [r , oo) x V that intersects the slice at right angles; for
example, take E = [r , oo) x L , where we consider Las contained in V.
We claim that the geodesic curvature k of the loop L inside E is identicallyequal to l. To see this, let T and N denote the unit tangent and unit outward
normal vector fields of L as a loop in E , respectively. Since E is normal to the
slice { r} x v) we have N = -8 I or. The geodesic curvature of L is defined byk = (DrN, T), where D denotes the induced covariant derivative on the surface
E. Let B denote a choice of unit normal vector field to E, so that {T, N , B} forms
an orthonormal frame along L. Since the second fundamental form II of the slice
{r} x V satisfies II= e-^2 r 9flat, we have^13
DrN = 'VrN - ('VrN, B) B =II (T) - II (T, B) B = T,
where 'V denotes the covariant derivative of the ambient cusp. Therefore k = 1 forthe loop Lin E.
Now since (MA (t) , g(t)) converges to (1iA, hl'HJ and since TA (t) is a torus
of constant mean curvature of fixed area A , for any E > 0, there exists t" (2) < oo
such that for to :::'.'. t" (2),
(33.56) ( kds(to)::::; (l+c)L 9 (to)(8V(to)),
J 8V(to)where L 9 denotes the length with respect tog.
l3 As a (1, 1)-tensor, II= id.