4. ANÁLISIS E INTERPRETACIÓN DE RESULTADOS
4.1. COMPROBACIÓN DE LA HIPÓTESIS
kernel of this action by H0. Choose a connected fundamental domain
Λ0for theρΛ–action, and letM0be the corresponding subset ofM−N.
Let Σ be the boundary spheres of M0 which are not contained in∂N
and choose aρ(H)–invariant spherical metric onρ(H)Σ. In the special case of Λ =S1 andρ
Λthe trivial action (H0=H) take
an arbitrary leafF, equip it with aρ–invariant spherical metric and let M0 ∼=S2×IbeM cut open alongF (withF added on both ends of the
interval).
Now∂M0 is equipped with a spherical metric, and it remains to ex-
tend this metric ρ(H0)–invariantly to the interior of M0. Since Λ0 is
a fundamental domain forρΛ, such a metric onM0 lifts to the desired
ρ(H)–invariant metric onM−N.
In order to construct aρ(H0)–invariant (S2×R)–metric onM0, note
that the foliation has aρ(H0)–invariant transversal line field. This in-
duces aρ(H0)–equivariant diffeomorphismφbetween the two boundary-
spheres Σ0 = S2 × {0} and Σ1 = S2× {1}. We take a fixed ρ(H0)–
equivariant isometry to identify Σ0 and Σ1 and apply Proposition 1.23
to conclude that the diffeomorphism φ: Σ0 → Σ1 is equivariantly iso-
topic to an isometric one. Using this isotopy, we equivariantly modify the line field such that the induced diffeomorphism gets an isometry. Now take the product metric onM0 with respect to the trivialization of
the modified line field, i. e. leaves are totally geodesic and orthogonal to the line-field.
3.2 Invariant foliation of the neck-like region
Let M be a (not necessarily complete) Riemannian manifold, and let ρ: G yM be an isometric finite group action. Recall that Mneck is the set of points inM, which are centers of–necks. Since the action is by isometries, Mneck
is ρ(G)–invariant. In this section we construct a ρ(G)–invariantS2–foliation ofMneck.
Definition 3.3. We call a (unit) tangent vectorv ∈ TpM at a point p∈Mneck
adistant direction, if there exists a minimizing geodesicγ of lengthS−12(p)1
3 starting fromγ(0) =pwith ˙γ(0) =v.
Definition 3.4. A smoothly embedded surface Σ⊆Mis calledν–hori- zontal, if angles between Σ and all distant directions are> π2 −ν.
3 Invariant singularS –foliations
Each neckη: S2(√2)×(−1
,
1
)→M defines aheight function hη= π(−1
, 1 )◦η
−1
on its image, whereπ(−1 ,
1
)is the projection on the interval factor. We call the level setsh−1
η (t) =η(S2× {t})leavesof the neckη. For an –neck η: S2 ×(−1
,
1
) → M define the inner half Vη = η(S2×(−1
2,
1
2)), theinner thirdWη=η(S
2×(−1 3,
1
3)), and theinner quarter Uη=η(S2×(−1
4,
1 4)).
To simplify notation, we use the convention that a 0–neckis an infinite round cylinder.
Proposition 3.5 (distant directions in–necks). There is a mono- tonically increasing functionθ1: [0,1001 )→[0,∞)withlim→0θ1() = 0
such that the following holds: Letη: S2(√2)×(−1
,
1
)→M be an–neck with <
1
100, then for any
p∈Vηthere are distant directionsv1,v2 with∠p(v1, v2)≥π−θ1()and
for any two distant directionsw1,w2 holds either∠p(w1, w2)≥π−θ1()
or∠p(w1, w2)≤θ1().
Proof. To simplify notation assume thatS(p) = 1, soηis an–isometry. We denoteη−1(p) by (x, t)∈S2(√2)×(−1 2, 1 2). Letq1=η((x, t−21)),q2=η((x, t+21)). Then ˛ ˛d(qi, p)−21 ˛ ˛≤1 and ˛ ˛d(q1, q2)−1 ˛
˛≤1. Therefore minimizing geodesicsγi(i= 1,2) fromp
toqihave length>31 and by triangle comparison∠p( ˙γ1(0),γ˙2(0))→π
for → 0 (note that curvature on im(η) is almost non-negative). We putvi:= γ˙i(0)
kγ˙i(0)k and choose a monotonically increasing functionθ1with lim→0θ1() = 0 such that∠p( ˙γ1(0),γ˙2(0))≥π−θ13().
On the other hand, assume thatwis any distant directions atp. Let γ be a geodesics starting frompwith ˙γ(0) =wand letq=γ(1
3). Then η−1(q) has distance at most 1 fromS2× {t− 1
3}or toS
2× {t+ 1 3}. From the two distant direction constructed above let vi be the one for whichγidoesnot get close toqand put ¯q:=γi(1
3). Then holds d(¯q, q)> 2
3−2 and d(p, q) =d(p,q) =¯
1 3
so as before for→0 we have∠p( ˙γi(0),γ(0))˙ →π, and we may modify θ1 such that∠p( ˙γi(0),γ(0))˙ ≥π−θ13(). This shows that∠p(w,−vi)≤
θ1()
3 , so any distant direction has angle≤
θ1()
3 with either−v1 or−v2.
This proves the proposition.
Proposition 3.6 (leaves almost horizontal). There is a monotoni- cally increasing functionθ2: [0,1001 )→[0,∞)withlim→0θ2() = 0such
3.2 Invariant foliation of the neck-like region
that the following holds: Let η be an –neck andΣ a leaf in the inner halfVη, i. e.Σ =h−η1(t)witht∈(−21,21). ThenΣisθ2()–horizontal.
Proof . Let γ be a minimizing geodesic of length 3 starting from p in any distant direction. Then η−1(γ) is a minimizing η∗g–geodesic in S2(√2)×(−1
,
1
), and it is uniquely minimizing between x:=η
−1(p)
andy:=η−1(γ(61)). By Proposition 1.16 the angle atxbetweenη−1(γ) and the minimizinggcyl–geodesicγcylfromxtoygoes to 0 for→0. But sinceγcylis minimizing of length≥ 1
8(with respect togcyl), it must be almost orthogonal onS2×{t}=η−1(Σ), with∠gcyl(η
−1
(Σ),γcyl(0))˙ →0 for→0. The claim follows now fromC1
–closeness ofgandgcyl.
Corollary 3.7. Let η1, η2: S2( √ 2)×(−1 , 1 )→M be –necks around xi that intersect. Then
1. if the inner half intersect, then atx∈Vη1∩Vη2 holds
∠(dhη1, dhη2)<2θ2() or ∠(dhη1, dhη2)> π−2θ2(). 2. if the inner thirds intersect and θ2() < 101, then any leaf Σ =
h−η11(t)ofη1 is isotopic to any leaf ofη2.
Proof . Let Σ1, Σ2 be the leaves through a point x ∈ Vη1 ∩Vη2, and choose a distant directions v ∈ TxM. By Proposition 3.6,v is almost orthogonal on both Σ1 and Σ2, thus the angle between Σ1and Σ2is less
2θ2(). This implies the first statement.
For the second statement, first isotope Σ to a leaf Σ0=h−η11(t
0
) inter- sectingWη1∩Wη2. Since the diameter of Σ
0
is close to√2πS(x1)−
1 2 by Proposition 3.6, Σ0lies in the inner halvesV1∩V2. Thus at every point
x∈Σ0, the angle between Σ0and theη2–leaf throughxis≤2θ2(). This
implies thatη2−1(Σ
0
) hits every{a} ×(−1
,
1
) exactly once, and therefore Σ0 can be isotoped to a leaf ofη2.
We now construct a globalρ(G)–invariant smoothS2–foliation on the unionUof all inner quarters of–necks,
W:= [ ηis–neck Wη ⊇ U:= [ ηis–neck Uη ⊇ Mneck .
Lemma 3.8. For givenKthere exists a monotonically increasing func- tion θ: [0, 1
100) → [0,∞) with lims→0θ(s) = 0 such that the following
3 Invariant singularS –foliations
Letρ: Gy(M, g) be an isometric finite group action with|G| ≤ K
on a complete connected Riemannian manifold (M, g) and let≤ 1 100.
Then there exists a ρ(G)–invariant smooth S2–foliationF
on an open setF,U⊆F⊆M with the following properties:
Ifxis center of an˜–neck for˜≤andΣis a leaf ofFthroughx, then Σisθ(˜)–horizontal. Furthermore, the foliationFonB(x,˜ θ(˜1))∩Uis θ(˜)–close inCθ(˜1)–topology to the metric product foliation ofS2(√2)×R. Proof . Of course, the ˜–necks η: S2 ×(−1
˜ , 1 ˜ ) → M give local S 2 – foliations, with almost horizontal leaves and the foliation ˜–close to the cylindrical standard foliation. The problem is that those approximations need not to beρ–invariant and different local foliations have to be glued together. In order to get a globalρ(G)–invariant foliation, the main idea is here to average the height functions (respectively its gradient) of a ρ(G)–invariant family of ˜–necks.
For eachx∈Mneckchoose ˜(x)∈[0, ] and an ˜(x)–neckηxsuch that ˜
(x) is almost as small as possible, i. e. any ˜0–neck around x satisfies ˜
0 > 23˜(x). (If ˜(x) = 0 for some point, then (M, g) is isometric to a round cylinder and the claim is trivial. So we may exclude this case.) Note that for anyyin the inner halvesVηx holds
3˜(x)≥˜(y)≥ 1
3˜(x), (3.2.1)
since ηx is also a 2˜–neck aroundy, and a 1
3˜(x)–neck aroundywould
be an 23˜(x)–neck aroundx, which cannot exist. Now from this family of necks choose a sub-family
Φ ={ηx: S2×(− 1 ˜ (x), 1 ˜ (x))→M}
of ˜(x)–necks such that their inner halves{Vηx|ηx∈Φ}form a locally finite covering ofWand choose a subordinate partition of unity{βη|η∈
Φ}. We may assume that the covering and the partition of unity isρ(G)– invariant, for if Φ is a given collection, than {ρ(g)η|η ∈ Φ, g ∈ G} is aρ(G)–invariant collection of–necks, and{ 1
|G|βη◦ρ(g) −1|
η∈Φ, g∈
G} is ρ(G)–invariant subordinate partition of unity. Since the regions Wηx are almost cylindrical, it is possible to bound the multiplicity of a (non-invariant) covering by 3 and thus the multiplicity of the invariant covering by 3|G|.
By Proposition 3.6 all leaves inVηxareθ2(˜(x))–horizontal, so kerdhηx
has angle≥π
3.2 Invariant foliation of the neck-like region
For anyy∈W we can choose signsy,η∈ {±1}such that by Corol- lary 3.7 and (3.2.1) holds forz neary
∠z(y,ηdhη(z), y,η0dhη0(z))<2θ2(3˜(y)) (3.2.2)
for all η, η0 with y ∈ Vη∩Vη0. Using these neck-orientations, we can
interpolate to define a local one-form neary α(z) :=X
η∈Φ
βη(z)y,ηdhη(z)∈Tz∗M
αis locally well-defined up to sign. Thus, we get a well-defined global one-form ˜αas a section in T∗M/±1 onW⊇U⊇Mneck. Note that ˜
αisρ(G)–invariant since Φ andβ are so.
Since small balls in the space of directions are convex, interpolation improves the horizontality, and so we conclude from Proposition 3.6 that kerαisθ2(3˜(y))–horizontal.
αhas a local primitivef neary, namely f(z) =X
η∈Φ
βη(z)y,ηhη(z).
So the (globally onWdefined) plane field ker ˜αis integrable and defines a foliationFby integral surfaces which are level sets off.
Note that on the overlapVηx∩Vηx0 the height functions areθ
0
–close in
C[θ10]–topology, withθ0→0 for max{˜(x),˜(x0)} →0. Thus, ifηis any ˜–
neck aroundy, the pulled back height functions and also the interpolated functionf is close to the cylindrical height function, and soFis close to the standard cylindrical foliation (in particular the leaves of F are spheres close to the ˜–neck-leaves). Using (3.2.1) and the bound on the multiplicity of the covering, one finds a functionθ(˜) as in the statement of the Lemma measuring the closeness ofFand the cylindrical foliation.
Since ˜αis defined onW and for eachx∈Uholds ˜B(x, 1
20)⊂W, we can takeFto be the union of all ker ˜α–leaves throughU.
We get an immediate consequence ifM is completely neck-like:
Corollary 3.9. SupposeM is a connected, orientable, closed 3–manifold and M = Mneck
with ≤ 1001 . Let ρ: Gy M be an isometric finite
3 Invariant singularS –foliations
Proof . Lemma 3.8 yields that M has a ρ(G)–invariant (smooth) S2– foliation. HenceM ∼=S2×S1, and the action is standard by Proposi- tion 3.2.
We finally note that invariant, almost horizontal spheres are isotopic to leaves ofF:
Lemma 3.10. Let≤ 1
100 be such that θ()< 1
10 andΣ⊂Uan em-
bedded 2–sphere that is 101–horizontal andρ(H)–invariant for a subgroup H ≤ G. Then Σ can be ρ(H)–equivariantly isotoped to a leaf of the foliationFfrom Lemma 3.8.
Proof. This is an equivariant version of the second claim in Corollary 3.7, and the proof is analogous: Since Σ is 1
10–horizontal and leaves ofFare
θ()–horizontal (withθ < 101), Σ intersects all leaves ofFtransversally. It follows that Σ hits each integral curve of ˜αexactly once. Since ˜αis invariant, the isotopy can now be doneρ(H)–equivariantly along these integral curves.