1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. INCOMPRESSIBILITY OF BOUNDARY TORI 239

THEOREM 33.25 (Meeks and Yau). Let (N^3 , g) be a compact Riemannian 3-

manifold with boundary. Suppose that fJN is convex, i.e., the second fundamental

form of fJN C N is positive with respect to the unit outward normal, and suppose

that fJN is compressible in N. Let J denote the space of all smooth maps f from

the unit disk D^2 into N such that flaD : fJD -+ fJN represents a nontrivial element

of 7ri ( fJN). Then there exists a conformal harmonic map f o : D -+ N in J of least

area among all maps in J. i^2 Moreover, Jo satisfies the following properties:

(1) The map Jo is an embedding and hence [folad is a primitive element of

ker { i* : 7ri ( fJN) -+ 7ri (N)}.

(2) For z E fJD, the tangent map dfo : TzD -+ Tfo(z)N takes the unit inner

normal of fJD to a nonzero inner normal to fJN (i.e., the embedded disk

Jo (D) is normal to fJN).

(3) If the boundary component containing Jo (D) is a torus T c fJN, then

[folaDl E 7ri (T) is a primitive element generating the infinite cyclic group

ker { i* : 7ri (T) -+ 7ri (N)} ~ Z.

REMARK 33.26. More generally (see p. 462 of [224]), if S c fJN consists of

some of the components of fJN, then in the space of all smooth maps f from the

unit disk D into N such that flaD : fJD -+ S represents a nontrivial element of

7ri ( S), provided this space is nonempty, there exists a conformal harmonic map

Jo: D-+ N which minimizes area.

Here we give a proof of part (3) of Theorem 33.25. We first show that the

homomorphism induced by inclusion

is not the zero map. Suppose on the contrary that j* is the zero map. Then by

attaching handlebodies to all boundary components of N other than T , we obtain

a compact orientable 3-manifold W^3 with boundary fJW = T. We have that the

map

k*: Hi (T,~)-+ Hi (W,~),

induced by the inclusion k : TY W, is also the zero map. This contradicts Lemma

31.20.

We claim that ker { i* : 7ri (T) -+ 7ri (N)} is not isomorphic to Z x Z. Otherwise,

by the well-known fact that a subgroup G of Z x Z with G ~ Z x Z has finite index,

it follows that ker (j) = ker (i) ® ~ = Hi (T, ~). This contradicts j* not being

the zero map.
Since a subgroup G of Z x Z is isomorphic to either 0, Z, or Z x Z, it follows that
ker {i : 7ri(T)-+ 7ri(N)} ~ Z. Now c = [folad is inker (i) and c is primitive.
Thus c is a generator of ker (i*). This completes the proof of part (3).

The rest of this section is devoted to the proof of Proposition 33.24.

4.2. Bounding the time derivative of the area of minimal disks in
terms of length and area.

In this subsection, as a first step in proving Proposition 33 .24, we shall derive
an upper bound for d+d~;; in terms of the length and area of a minimal disk (see
inequality (33.58) below).

(^12) By conformal harmonic map we mean a harmonic map which is a lso a conformal map.