Parabolic Foliations on Three-Manifolds
We prove that every closed orientable three-manifold admits a parabolic foliation.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1065392016-10-01T03:01:52Z Parabolic Foliations on Three-Manifolds Krouglov, V. We prove that every closed orientable three-manifold admits a parabolic foliation. 2009 Article Parabolic Foliations on Three-Manifolds / V. Krouglov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 170-191. — Бібліогр.: 5 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106539 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We prove that every closed orientable three-manifold admits a parabolic foliation. |
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Krouglov, V. |
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Krouglov, V. Parabolic Foliations on Three-Manifolds Журнал математической физики, анализа, геометрии |
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Krouglov, V. |
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Krouglov, V. |
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Parabolic Foliations on Three-Manifolds |
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Parabolic Foliations on Three-Manifolds |
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Parabolic Foliations on Three-Manifolds |
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Parabolic Foliations on Three-Manifolds |
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Parabolic Foliations on Three-Manifolds |
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parabolic foliations on three-manifolds |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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http://dspace.nbuv.gov.ua/handle/123456789/106539 |
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Parabolic Foliations on Three-Manifolds / V. Krouglov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 170-191. — Бібліогр.: 5 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT krouglovv parabolicfoliationsonthreemanifolds |
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2025-07-07T18:37:19Z |
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2025-07-07T18:37:19Z |
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1837014387629490176 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 2, pp. 170�191
Parabolic Foliations on Three-Manifolds
V. Krouglov
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:vkrouglov@gmail.com
Received October 10, 2008
We prove that every closed orientable three-manifold admits a parabolic
foliation.
Key words: parabolic foliation, extrinsic curvature.
Mathematics Subject Classi�cation 2000: 53C12, 57R30.
1. Introduction
It is well known that every closed orientable three-manifold admits a foliation.
When there are additional restrictions on the geometry and topology of the leaves,
this statement is not true. For example, the foliations by minimal surfaces do not
exist on a three-sphere (relatively to any metric). Analogously, the classes of
totally umbilical foliations and totally geodesic foliations do not exist on every
three-manifold.
A. Borisenko introduced new classes of foliations on the Riemannian mani-
folds having the restrictions on the extrinsic geometry of leaves, namely, elliptic,
parabolic, and strong saddle (or hyperbolic) foliations.
De�nition 1.1 (Borisenko). A codimension one foliation on a three-manifold
is called:
1) parabolic, if there is a metric such that Ke = 0;
2) (strong)saddle, if there is a metric such that (Ke < 0)Ke � 0;
3) elliptic, if there is a metric such that Ke > 0,
where Ke states for the extrinsic curvature of the leaves.
The studying of the existence of these foliations on three-manifolds was ini-
tiated by D. Bolotov in [2] where he, among other results, de�ned a metric on
c
V. Krouglov, 2009
Parabolic Foliations on Three-Manifolds
the solid torus such that the Reeb component was a parabolic foliation. In [3] he
gave the examples of strong saddle foliations on the torus bundles over the circle
and on a three-sphere. In particular, a foliation in the Reeb component is not
a topological restriction to the existence of strong saddle foliations. In [5], the
author showed that in fact every closed orientable three-manifold admits a strong
saddle foliation.
It is well known that closed orientable three-manifolds do not admit elliptic
foliations. Namely, the existence of them contradicts to a well-known Reeb for-
mula
R
M
H = 0 (here H stands for the mean curvature) since, if it is elliptic with
respect to some metric, then its total mean curvature cannot be zero.
The last open problem was the existence of parabolic foliations on closed
orientable three-manifolds. In this paper we give the positive answer to this
question.
Theorem 1.2. Every closed orientable three-manifold admits a parabolic foli-
ation.
Notice that there are no parabolic foliations on S3 with respect to a standard
metric.
Parabolic foliations of the codimension larger than one were studied in [1].
The paper is organized as follows. In Section 2 we recall some de�nitions and
constructions from the topology of foliations on three-manifolds. In Section 3,
several local models of parabolic foliations are constructed. In Section 4 we de�ne
a parabolic foliation on the three-sphere which is a turbulization of the Reeb
foliation along an arbitrary knot. In Section 5 it is shown how to perform a Dehn
surgery on this knot to obtain a parabolic foliation on every closed orientable
three-manifold.
Acknowledgements. The author is grateful to Dmitry Bolotov for his con-
stant support and attention paid to this work and to Prof. Alexander Borisenko
for his help and valuable advices.
2. Basic De�nition
2.1. Foliations on Three-Manifolds
In this section we recall some necessary basic facts about the foliations on
three-manifolds.
Let F be a foliation on a closed three-manifold. It de�nes a two-dimensional
distribution of planes tangent to the leaves. However, not every plane distribu-
tion de�nes a foliation. A distribution is called integrable if it de�nes a foliation.
Classical Frobenius theorem gives necessary and su�cient conditions for the dis-
tribution to be integrable. We also recall a three-dimensional version of this
theorem.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 171
V. Krouglov
Theorem 2.1 (Frobenius). A distribution of planes � on a three-manifold
is integrable if and only if for every pair of local sections X and Y of � its Lie
bracket belongs to �.
Notice that a distribution is called transversally orientable if there is a glob-
ally de�ned vector �eld that is transverse to it. In this case there is a globally
de�ned one-form � such that Ker(�)p = TpL, where L is a leaf through p. It is
easy to rewrite the conditions of Frobenius theorem in the terms of the form �:
a distribution is integrable if and only if � ^ d� = 0.
Example 2.2. Reeb foliation on D2 � S1.
Consider the following C1-smooth function on [0; 1]:
1) the function f is a smooth increasing function on [0; 1];
2) there is an " > 0 such that for any x 2 [0; ") the value of f(x) is equal to
zero and f(x) = 1.
On the solid torus D2 � S1 with cylindrical coordinates ((r; �); t) de�ne the fol-
lowing one-form:
� = f(r)dr + (1� f(r))dt:
By the Frobenius theorem, a distribution of planes de�ned by the kernel of � is
integrable since
� ^ d� = (f(r)dr + (1� f(r))dt) ^ (�f 0(r)dr ^ dt) = 0:
Therefore � de�nes a foliation on D2 � S1. We denote this foliation by FR and
call it the Reeb foliation on a solid torus.
Remark 2.2. The Reeb foliation is usually de�ned as a foliation of D2�S1 =
f((r; �); t) : r 2 [0; 1], �, t 2 [0; 2�)g by the levels of function h(r; �; t) =
(r2�1)et. It is obvious that the foliation de�ned above is isotopic to this foliation
what justi�es the title of Example 2.2.
2.2. Extrinsic Geometry of Foliations
Assume now that M is a Riemannian manifold with a scalar product g and
an associated Levi�Civita connection r. Consider a foliation F on M . For each
pair of vector �elds X and Y on M that are tangent to F , de�ne the second
fundamental form of F with respect to unit normal n by
B(X;Y ) = g(rXY; n):
Using the scalar product in the tangent bundle, we may de�ne the following linear
operator An:
B(X;Y ) = g(AnX;Y );
172 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
that is called a Weingarten operator. Since An is symmetric, it has two real
eigenvalues which are principal curvature functions. A product Ke = k1k2 is
called an extrinsic curvature of F .
Owing to the sign of extrinsic curvature we may de�ne the following classes
of foliations on three-manifolds.
De�nition 2.3 (Borisenko). A codimension one foliation on a three-manifold
is called:
1) parabolic, if there is a metric such that Ke = 0;
2) (strong)saddle, if there is a metric such that (Ke < 0)Ke � 0;
3) elliptic, if there is a metric such that Ke > 0.
Remark 2.4. As mentioned in the introduction, there are no elliptic foliations
on the closed oriented three-manifolds. Since Ke > 0, the functions of principal
curvatures are nowhere zero and simultaneously they are larger or less than zero.
By the Reeb formula
0 =
Z
M
H =
1
2
Z
M
(k1 + k2) 6= 0;
what is a contradiction.
Remark 2.5. Notice that many geometric classes of foliations are subclasses
of the introduced classes. Minimal foliations are saddle, totally umbilical foliations
have Ke � 0, and totally geodesic foliations are parabolic.
2.3. Knots and Braids
Recall that a knot in S3 or R3 is an image of a circle S1 under some C1-
smooth regular embedding. The knots K0 and K1 are called isotopic if there is
a smooth family of embeddings K(t) : S1 ! S3(R3) such that K(0) = K0 and
K(1) = K1.
Consider two sets of the points A = f(i; 0; 0); i = 1; : : : ; ng and B = f(i; 0; 1);
i = 1; : : : ; ng in R
3 . A smooth embedded curve
(t) is called descending if its
z-coordinate is a strictly decreasing function of the parameter t.
A topological braid K with n strings is a collection of n disjoint descending
curves in R3 which connect the points from the set B with the points of A. We
say that two braids are isotopic if there is a smooth family of braids connecting
them.
Consider a group with generators �1; �2; : : : ; �n�1 and relations �i�i+1�i =
�i+1�i�i+1 for all i and also �i�j = �j�i in the case when ji� jj � 2. This group
is denoted by Bn and called a group of algebraic braids. There is a one-to-one
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 173
V. Krouglov
(i - 1)
i(i - 1)
i
A
B
(i - 1)
i(i - 1)
i
A
B
Fig. 1: Possible intersections in the frontal projection.
correspondence between the isotopy classes of topological braids and the elements
of Bn. Further we will agree that a topological braid whose frontal projection is
in the left part of Figure 1 corresponds to the generator �i. The second possi-
ble intersection corresponds to the element ��1
i
. Therefore each isotopy class of
topological braids can be represented as a product K = ��11 ��12 : : : ��1
N
.
On the set of isotopy classes of topological braids de�ne a new operation �
a closure of the braid. A closure of braid K is a link (that is an embedded image
of disjoint S1) which is obtained from K by adding disjoint curves connecting the
i-th point of A with the i-th point of B. The following theorem holds (cf. [4]):
Theorem 2.6. The mapping from the set of isotopy classes of topological braids
which maps each braid to its closure is surjective. In particular, each isotopy class
of knots contains the closure of some braid.
2.4. Combinatorial Presentation of Three-Manifolds
In this section we will give a sketch of the proof that every closed orientable
three-manifold admits a foliation.
Consider a knot K in S3. Let N be some tubular neighborhood of K. Denote
X = S3nN . Then @X = @N = T 2. Consider some homeomorphism
h : @X ! @(D2 � S1);
and let M = X [D2� S1=(y � h(y); for all y 2 @X). It is easy to see that M is
a closed manifold.
This construction is called a Dehn surgery on a knot. The importance of this
construction follows from the theorem:
Theorem 2.7. [4] Any closed orientable manifold may be obtained by the Dehn
surgery on some knot in S3.
174 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
Recall a classical construction of the transversally orientable foliation on
a closed orientable three-manifold. Consider a solid torus D2 � S1 = f((r; �); t) :
r 2 [0; 2]; �; t 2 [0; 2�)g, and let � = f(r)dr + (1 � f(r))dt, where f(r) is some
smooth function on the segment [0; 2] satisfying the following conditions:
1) f(r) is a strictly increasing function on [0; 1];
2) there is an � such that for all r 2 (2� �; 2] the function f(r) = 0;
3) f(r) is a strictly decreasing function on [1; 2� �];
4) f(0) = 0 and f(1) = 1.
The form � de�nes some foliation on D2 � S1. Denote it by FT .
It is obvious that FT has a single compact leaf fr = 1g. FT , restricted on
a solid torus D2(1) � S1 = f(r; �; t) 2 D2 � S1 : r 2 [0; 1]g, is a Reeb foliation
(see Example 2:2).
It is well known that S3 may be represented as a union of two solid tori
that are glued along the boundary torus. Gluing homeomorphism interchanges
the generators of the boundary torus. In each solid torus consider the Reeb
foliations FR. Since the gluing homeomorphism maps a leaf of the �rst Reeb
component to the leaf of the second one, we can see that a three-sphere admits
the foliation which is the union of two Reeb components. We will also denote this
foliation by FR.
Assume now that K is a knot in S3. By Theorem 2:4 it is isotopic to the
closure of some braid. Further we can isotope this braid to make it everywhere
transverse to the foliation of one of solid torus by disks D2 � ftg. Since FR is
a foliation by disks in a small neighborhood of the core curve r = 0, we may
assume that K is transverse to FR. Cut out a small tubular neighborhood of K
and glue it back into a solid torus with the foliation FT inside. We obtain a new
foliation on S3 which is a turbulization of the initial one along K. Finally, to
obtain a foliation on M , we Cut out a tubular neighborhood of K up to the torus
leaf and glue it back by di�eomorphism of the boundary. It is easy to verify that,
since the boundary of this neighborhood is a leaf, the foliation is correctly de�ned
on M . By Theorem 2:7 any closed orientable three-manifold may be obtained as
above, therefore every closed orientable three-manifold admits a foliation.
2.5. Bump Functions on R
Later in the proof we will often meet the situation when in some �nite segment
[a; b] a smooth function f(t) is de�ned in such a way that the following conditions
are satis�ed:
1) f(a) = f0;
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 175
V. Krouglov
2) f(b) = f1;
3) there is an � > 0 such that:
� for all t 2 [a; a+ �), f(t) = f0;
� for all t 2 [b� �; b), f(t) = f1;
4) f is monotone on [a; b].
In the paper we refer to these functions as to the bump functions on [a; b].
For example, in the construction of Reeb component FR, f is an increasing
bump function on [0; 1]. The function f(r), arising in the construction of FT , is
a union of two bump functions.
� �
�
Fig. 2: A graph of f(r) in the construction of turbulization.
��
�
Fig. 3: A typical bump function on [a; b].
176 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
3. Local Models of Parabolic Foliations
In this section we describe several local models of parabolic foliations on three-
manifolds.
3.1. Parabolic Foliation on �2 � [0; 1]
Lemma 3.1. Let �2 be a compact parallelizable surface (possibly with the
boundary). Let two Riemannian metrics G and H on �2 coincide in some neigh-
borhood of the boundary @�2. Assume that F is a foliation of M = �2 � [0; 1] by
the surfaces �2 � ftg. Then, there is such Riemannian metric g on M that:
1) in some tubular neighborhood of �2 � f0g, g = dt2 +G(p), for all p 2 �2;
2) in some tubular neighborhood of �2 � f1g, g = dt2 +H(p), for all p 2 �2;
3) F is parabolic on �2 � [0; 1] with respect to g;
4) there is a neighborhood U of the boundary @�2 such that for all t 2 [0; 1],
g(p; t)jU�ftg = G(p).
P r o o f. Let fX0; Y0g be an orthonormal frame on �2 with respect to H.
The matrix of G in this frame may be written as
G = G(p) =
�
a(p) b(p)
b(p) c(p)
�
for all p 2 �2. Since �2 is compact, the functions a, b and c are bounded on �2.
Consider a frame (X;Y; n) onM , where X = (X0; 0), Y = (Y0; 0), and n = @
@t
.
Let N be such a neighborhood of the boundary @�2 that GjN = HjN . Denote
L =MnN .
We are going to interpolate from G to H on �2� [0; 1] by using the following
Riemannian metric:
g = g(p; t) =
0
@ a(p; t) b(p; t) 0
b(p; t) c(p; t) 0
0 0 1
1
A ;
where (p; t) 2 �2 � [0; 1], a(p; t), b(p; t) and c(p; t) are functions on �2 � [0; 1].
The matrix of g is written with respect to the frame (X;Y; n).
By the de�nition of g, n is a unit normal vector �eld to F . Calculate the
matrix of the second fundamental form of the leaves relative to the normal n in
the basis fX;Y g. By the Koszul formula,
2g(rXX;n) = 2X(g(X;n)) � n(g(X;X)) + g([X;X]; n) � 2g([X;n];X);
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 177
V. Krouglov
where r is a Levi�Civita connection of g. Since X is independent of t, then
g([X;n];X) = 0. Therefore g(rXX;n) = �1
2
@a
@t
.
Similarly,
2g(rY Y; n) = 2Y (g(Y; n)) � n(g(Y; Y )) + g([Y; Y ]; n)� 2g([Y; n]; Y ) = �ng(Y; Y ):
Finally, we have
2g(rXY; n) = X(g(Y; n)) + Y (g(X;n)) � n(g(X;Y )) + g([X;Y ]; n)� g([Y; n];X)
�g([X;n]; Y ) = �n(g(X;Y )) + g([X;Y ]; n)� g([Y; n];X) � g([X;n]; Y ):
Since F is a foliation, we have g([X;Y ]; n) = 0. As X and Y do not depend
on t, we see that g([Y; n];X) = 0 and g([X;n]; Y ) = 0. Therefore g(rXY; n) =
�1
2
n(g(X;Y )). Consequently, the second fundamental form of the leaves is given
by the matrix
B = �
1
2
@a
@t
@b
@t
@b
@t
@c
@t
!
:
An extrinsic curvature of F with respect to g equals
Ke =
1
4
@a
@t
@c
@t
� @b
@t
2
ac� b2
:
Let 0 � t1 < t2 < t3 < t4 < t5 � 1 be a subdivision of the segment [0; 1]. Assume
that D is a positive real number larger than maxp2�2fa(p); c(p)g. We will choose
the exact value of D later in the proof.
Consider the following function h on �2:
1) h(p) = 1, for all p 2 L;
2) h(p) = 0 in some neighborhood of @�2;
3) h is a smooth nonnegative function on �2.
Consider the function ~a(p; t) = Df(t)+(1�f(t))a(p), where f(t) is an increasing
bump function on [0; t1] with f(0) = 0 and f(t1) = 1. Finally, let
a(p; t) = h(p)~a(p; t) + (1� h(p))a(p):
On [0; t1] de�ne the following matrix (with respect to the frame (X;Y; n)):
gD = gD(p; t) =
0
@ a(p; t) b(p) 0
b(p) c(p) 0
0 0 1
1
A :
178 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
This matrix clearly de�nes a metric on �2 � [0; t1] since, when t = 0, it has
positive diagonal entries, and a(p; t) is nondecreasing on the segment [0; t1]. By the
de�nition of bump functions, g is equal to dt2+G(p) in some tubular neighborhood
of �2 � f0g. A foliation by surfaces � � ftg is parabolic with respect to the
introduced metric.
On the segment [t1; t2] we may change c(p) in the same way. Consequently,
on the segment [0; t2] we have:
1) gD(p; 0) =
�
G(p) 0
0 1
�
;
2) gD(p; t2) =
0
@ a(p; t2) b(p) 0
b(p) c(p; t2) 0
0 0 1
1
A;
3) F is a parabolic foliation on �2 � [0; t2] with respect to gD.
Consider an increasing bump function f(t) on the segment [t2; t3] with f(t2) = 0
and f(t3) = 1. On [t2; t3] de�ne gD by the matrix
gD =
0
@ a(p; t2) + f(t)b(p) b(p)(1 � f(t)) 0
b(p)(1 � f(t)) c(p; t2) + f(t)b(p) 0
0 0 1
1
A :
Since b(p) is bounded on �2 and equal to zero on N , the matrix of gD is positively
de�nite for some choice of D (see the de�nition of D above) and therefore de�nes
a metric.
The foliation is parabolic with respect to the introduced metric since an ex-
trinsic curvature is given by Ke =
1
4det(gD)
(f 0(t)2b(p)2 � (�f 0(t))2b(p)2) = 0.
On �2 � ft3g the matrix of gD is diagonal. All non-diagonal elements of gD
being eliminated, we may freely decrease the diagonal elements of gD. To do
this, consider the decreasing bump function k(t) on [t3; t4] with k(t3) = 1 and
k(t4) = 0. On [t3; t4] de�ne
gD =
0
@ k(t)a(p; t3) + (1� k(t)) 0 0
0 c(p; t3) 0
0 0 1
1
A :
For all t 2 [t3; t4] the matrix gD is positively de�nite and therefore de�nes a metric.
At t = t4 the matrix of gD is given by
gD =
0
@ 1 0 0
0 c(p; t3) 0
0 0 1
1
A :
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 179
V. Krouglov
Analogously, on [t4; t5] we may decrease a diagonal element c(p; t3). We showed
above how to deform gD into the metric
0
@ 1 0 0
0 1 0
0 0 1
1
A preserving the parabolicity
de�ned of F . This metric is nothing else but a matrix of dt2+H(p) written with
respect to a frame (X;Y; n).
Let U be an open subset of fp 2 �2;where h(p) = 0g containing the boundary
of �2. On [0; t1] the function a(p; t)jU = a(p)jU = 1. The same holds for the
function c(p; t)jU on the segment [t1; t2]. On [t2; t3], since b(p)jU = 0, the matrix
of gjU is an identity matrix. On the segments [t3; t4] and [t4; t5], when a(p; t)jU =
c(p; t)jU = 1, the matrix gjU is also an identity matrix. This �nishes the proof of
the lemma.
Corollary 3.2. Consider the manifold M = T 2 � [0; 1]. Let G(x; y) and
H(x; y) be some metrics on T 2. Then there is a metric g on M such that:
1) foliation F by the tori T 2 � fptg is parabolic;
2) the matrix of gjT 2�f0g =
�
G(x; y) 0
0 1
�
and the matrix of gjT 2�f1g =�
H(x; y) 0
0 1
�
;
3) g is a direct product metric in some one-sided neighborhood of the boundary.
3.2. Parabolic Foliation on a Solid Torus
The following proposition is due to D. Bolotov.
Lemma 3.3 (D. Bolotov, [2]). There is a foliation F and a metric g on
D2 � S1 such that:
1) F is parabolic with respect to g;
2) the foliation Fj
D2( 1
3
)�S1 is a foliation by the totally geodesic disks D2(1
3
)�ftg
and the foliation Fj([ 2
3
;1]�S1)�S1 is a foliation by the totally geodesic tori
frg � S1 � S1.
P r o o f. Consider the solid torus D2�S1 with the following coordinates on
it:
D2 � S1 = f((r; �); t) : r 2 [0; 1]; �; t 2 [0; 2�)g:
De�ne the one-from � on D2 � S1 as
� = f(r)dr + (1� f(r))dt;
180 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
where f(r) is such a smooth function on [0; 1] that
f(r) =
8<
:
0; r 2 [0; 1
3
]
is a strictly increasing function when r 2 (1
3
; 2
3
]
1; r 2 (2
3
; 1]
:
This form de�nes a "thick" Reeb foliation F on D2�S1 (that is, there is a subset
N such that FjN is a Reeb foliation and FjD2�S1nN is di�eomorphic to a product
foliation by tori).
Assume that in the coordinates (r; �; t) the matrix of g has a form
g =
0
@ 1 0 0
0 G(r) 0
0 0 1
1
A :
In order to calculate the second fundamental form of F consider the following
sections: X = @
@�
; Y = (1 � f(r)) @
@r
� f(r) @
@t
of the tangent bundle TF . Let
n = f(r) @
@r
+ (1� f(r)) @
@t
be a normal vector �eld.
By a direct computation we obtain a matrix of the second fundamental form
that is equal to
1
2f(r)2 � 2f(r) + 1
�
�f @G
@r
0
0 �(1� f)@f
@r
�
:
It is obvious, since f = 0 on [0; 1
3
), the foliation by disks is totally geodesic for
every choice of G = G(r). De�ne G = G(r) in the following way:
G =
8<
:
r2when r 2 [0; 1
4
)
strictly increasing when r 2 [1
4
; 1
3
)
1 when r 2 [1
3
; 1]
:
For this choice of G, the metric g is regular in the neighborhood of the core curve
r = 0 and satis�es the conditions of the lemma.
3.3. Parabolic Foliation on T 2 � [0; 1]
Using the similar arguments, we may obtain the following result:
Lemma 3.4. There exists a foliation F and a metric g on T 2 � [0; 1] such
that:
1) F is parabolic with respect to g;
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 181
V. Krouglov
2) the foliation Fj
T 2�[0; 1
3
] is a foliation by totally geodesic tori T 2 � frg and
the foliation Fj
S1�S1�[ 2
3
;1] is a foliation by totally geodesic annuli ftg�S1�
[2
3
; 1].
P r o o f. On T 2 � [0; 1] de�ne the following coordinates:
T 2 � [0; 1] = f((�; t); r) : r 2 [1; 2]; �; t 2 [0; 2�)g
and consider the one-from �
� = f(r)dr + (1� f(r))dt;
where f(r) is such a smooth function on [0; 1] that
f(r) =
8<
:
1; r 2 [0; 1
3
]
strictly decreasing, when r 2 (1
3
; 2
3
]
0; r 2 (2
3
; 1]
:
This form de�nes a foliation F on T 2�[0; 1]. Analogously as in the proof of Lemma
3:3 we may de�ne a matrix of g (with respect to the coordinates ((�; t); r)) in the
form
g =
0
@ G(r) 0 0
0 1 0
0 0 1
1
A :
To calculate the second fundamental form of the leaves consider the following
sections of TF : X = @
@�
; Y = (1� f(r)) @
@r
� f(r) @
@t
. The �eld n = f(r) @
@r
+ (1�
f(r)) @
@t
is a normal vector �eld.
The second fundamental form of F with respect to the unit normal n
jnj
is given
by
1
2f(r)2 � 2f(r) + 1
�
�f @G
@r
0
0 �(1� f)@f
@r
�
:
Since f = 0 on (2
3
; 1],it is obvious that the foliation F
T 2�( 2
3
;1] by horizontal annuli
is totally geodesic for an arbitrary choice of G(r). De�ne G = G(r) by the
following formula:
G =
8>><
>>:
1;when r 2 [0; 2
3
)
strictly decreasing, when r 2 [2
3
; 3
4
)
strictly increasing, when r 2 [3
4
; 4
5
)
r2, when r 2 [4
5
; 1].
For this choice of G, a metric g satis�es all conditions of the lemma.
182 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
Fig. 4: The function G(r) in the construction of a metric on T 2 � [0; 1].
4. Parabolic Foliations on Three-Sphere
The aim of this section is to de�ne a parabolic turbulization of F on S3 along
a knot K. We de�ne this foliation in several steps. First, we de�ne a parabolic
Reeb foliation on S3. Then we construct a special parabolic turbulization of FR
along the trivial link consisting of n components. For the knot K we consider its
special presentation which coincides with the trivial link everywhere except double
points in the frontal projection. Thus we de�ne a parabolic foliation everywhere
except some balls in S3 containing these double points. To de�ne a parabolic
foliation inside these balls we "twist" the turbulization along a trivial link with
two components. Further we will show how to glue these foliations back into the
sphere to get the desired foliation.
4.1. Parabolic Reeb Foliation on S3
Proposition 4.1 [2]. A three-sphere admits a parabolic foliation.
P r o o f. Take some presentation of the three-sphere S3 = D2
1�S
1[hD
2
2�S
1
as a union of two solid tori. De�ne the parabolic foliations F1 and F2 inside these
solid tori as in Lemma 3:3. In the coordinates (t; �) on @D2 � S1 the gluing
di�eomorphism h is given by the matrix h =
�
0 1
�1 0
�
. It is obvious that h is
an isometry of the boundary torus @(D2
1�S
1). Since the metrics on the solid tori
are direct product metrics and the foliations are direct product foliations in the
(one-sided) neighborhoods of boundary torus, then there is a well-de�ned glued
foliation FR and the metric on S3. This foliation is parabolic with respect to the
glued metric.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 183
V. Krouglov
4.2. Parabolic Turbulization along the Trivial Link
Proposition 4.2. For every n 2 N there is a parabolic foliation on S3 with n
"thick" Reeb components inside the solid torus D2(1
3
)� S1 � D2
1 � S1 � S3.
P r o o f. Consider a foliation FR on S3 de�ned as in Proposition 4:1. Notice
that the metric inside disk D2(1
3
) � f0g is a standard euclidian metric. Let
fx1; x2; : : : ; xng be a set of vertices of the regular polygon lying inside D2(1
3
)�f0g
with the center at r = 0 and the radius of the circumscribed circle equal to 1
8
with
respect to the metric induced on the disk D2(1
3
)�f0g. Instead of the radius 1
8
we
may choose any other one such that the circle with the center at the median of
the side xixi+1 and the radius equal to its length would entirely lie inside the disk
D2(1
3
). Take such " that any two circles with the centers at vertices of polygon
and of radius " are disjoint. Consider a set of vertical circles fxi � S1g passing
through the vertices xi.
On the solid torus D2 � S1 take the following coordinates:
D2 � S1 = f((r; �); t) : r 2 [0; "]; �; t 2 [0; 2�)g
and consider the function f given by the formula
f(r) =
8>>>><
>>>>:
0;when r 2 [0; "
6
];
strictly increasing, when r 2 ( "
6
; "
3
];
1;when r 2 ( "
3
; 2"
3
];
strictly decreasing, when r 2 (2"
3
; 5"
6
];
0;when r 2 (5"
6
; "]:
The one-form � = f(r)dr + (1 � f(r))dt de�nes a foliation F 0 on D2 � S1.
To de�ne a metric on D2�S1 consider the following function G = G(r) on it:
G =
8>>>><
>>>>:
r2;when r 2 [0; "
8
);
strictly increasing, when r 2 [ "
8
; "
6
);
�, r 2 [ "
6
; 5"
6
);
strictly increasing, when r 2 [5"
6
; 6"
7
];
r2, when r 2 [6"
7
; "].
De�ne a Riemannian metric g on D2 � S1 by the following matrix:
g =
0
@ 1 0 0
0 G(r) 0
0 0 1
1
A :
It is easy to verify that F 0 is parabolic with respect to this metric.
Cut out the �-tubular neighborhoods of the circles fxig � S1 and glue the
solid tori (D2 � S1;F 0) instead of these neighborhoods by identity map. Since
184 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
Fig. 5: Construction of metric on D2 � S1.
F 0j
fT 2�[ 5�
3
;2�]g is a foliation by totally geodesic annuli, it glues correctly to the
foliation of D2(1
3
)� S1 by horizontal disks.
By the construction, the metric on each D2 � S1 may be extended by the
euclidian metric on D2(1
3
) � S1 de�ned in Lemma 4:1. Denote the obtained
foliation by Fn.
De�nition 4.3. We call thus de�ned foliation Fn a trivial turbulization with
n strings.
4.3. Standard Presentation of a Knot
Assume that K is a knot in S3. We may isotope it in such a way that K is
a closure of some braid lying inside the solid torus D2(1
3
)�S1 and it is transverse
to a foliation of D2(1
3
)� S1 by totally geodesic disks. Write a presentation of K
as a product of transpositions K = ��11 ��12 : : : ��1
N
.
Without loss of generality, we may assume that in the frontal projection
f : D2 � S1 ! [�
1
3
;
1
3
]� S1 f(x; y; t) = (x; t)
there is a �nite number of levels t1; t2; : : : ; tN 2 [�1
3
; 1
3
] such that at these points
K has transverse double points (each point corresponds to transposition). Now
we can isotope K so that it becomes a subset of
S
n
k=1fxk � S1g for some n,
maybe except the neighborhoods of inverse images of double points f�1(ti); i =
1; 2; : : : ; N .
De�nition 4.4. We call such a presentation of K a standard presentation of
a knot with n strings.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 185
V. Krouglov
Fig. 6: Trivial turbulization with 5 strings (left). Standard presentation
of the knot (right).
The trivial turbulization Fn with n strings coincides with the turbulization
along a standard presentation of K everywhere except some balls around the
inverse images of double points.
4.4. Parabolic Foliation in the Neighborhood of Transposition
To de�ne turbulizations along transpositions consider a trivial turbulization
F2 with two strings on D2(1
3
)� [0; 1] (here we slightly abuse the notation and call
by trivial turbulization the foliation induced on D2(1
3
) � [0; 1] = D2 � S1nD2).
The metric, where F2 is parabolic, denote by g de�ned in Proposition 4:2. Let Æ
be some small enough real number. De�ne the following bump function f = f(r)
on [0; 1
3
]:
f(r) =
8<
:
�; r 2 [0; 1
4
+ Æ0]
strictly decreasing, when r 2 (1
4
+ Æ0;
1
3
� Æ]
0; r 2 (1
3
� Æ; 1
3
]
for some su�ciently small Æ0.
Consider a one-dimensional dynamical system (D2(1
3
); t) generated by the
�ow of the vector �eld X = f(r) @
@�
on the disk D2(1
3
) = f(r; �) : r 2 [0; 1
3
]; � 2
[0; 2�)g.
Let h(t) be a smooth increasing bump function on [0; 1] such that h(0) = 0
and h(1) = 1. Associate with it the following di�eomorphism � of D2 � [0; 1]:
�(r; �; t) = ( h(t)(r; �); t):
186 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
d
Fig. 7: A dynamical system used to de�ne the left parabolic transposition.
This di�eomorphism de�nes a foliation �(F2) on D
2�[0; 1]. It is clearly parabolic
with respect to the pull-back metric (��1)�g.
Assume that G(r; �) is a metric induced on the disk D2(1
3
)�f1g by metric g,
and H(r; �) is the metric induced on this disk by (��1)�g. We may use Lemma
3:1 to interpolate between these two metrics.
Notice that G(r; �) = H(r; �) on a disk D2(1
4
) � f1g since F2 is invariant
under the rotation by �. It is also clear that G(r; �) = H(r; �) on (D2(1
3
)nD2(1
3
�
Æ)) � f1g.
Let N = D2(1
3
� Æ + Æ0)nD
2(1
4
)� [0; 1]. By Lemma 3:1 there is such a metric
g on N that:
1) in some tubular neighborhood of (D2(1
3
� Æ + Æ0)nD
2(1
4
)) � f0g the metric
g(p; t) = G(p) + dt2 for all p 2 D2(1
3
� Æ + Æ0)nD
2(1
4
);
2) in some tubular neighborhood of (D2(1
3
� Æ + Æ0)nD
2(1
4
)) � f1g the metric
g(p; t) = H(p) + dt2 for all p 2 D2(1
3
� Æ + Æ0)nD
2(1
4
);
3) F is parabolic on N with respect to g;
4) There is a neighborhood U of the boundary @(D2(1
3
� Æ + Æ0)nD
2(1
4
)) such
that for all t 2 [0; 1]; g(p; t) = G(p) + dt2.
On (D2(1
3
)nD2(1
3
� Æ)) � [1; 2] and D2(1
4
) � [1; 2] consider the direct product
foliations. They are parabolic (even totally geodesic) with respect to the metric
ds2 = dr2+ r2d�2+ dt2. Since in the neighborhood of boundary @N the foliation
is a direct product foliation and the metric is a direct product metric, then there
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 187
V. Krouglov
Fig. 8: A trivial parabolic transposition with two strings (left). The left parabolic
transposition (right).
is a parabolic foliation correctly de�ned on the union L = (D2(1
3
)nD2(1
3
� Æ)) �
[1; 2] [N [D2(1
4
)� [1; 2].
Finally, consider the gluing �(F2) [ L. It is obvious that the foliations and
the metrics on L and �(F2) are smoothly glued with each other and they de�ne
the structure of parabolic foliation on the union. We are remained to "normalize"
this foliation in the t direction. For this consider the map
F : D2 � [0; 1]! �(F2) [ L
de�ned by the formula F ((r; �); t) = ((r; �); 2t). A foliation formed by inverse
images of the leaves is parabolic in the pull-back metric F �g.
We call the foliation obtained (together with the Riemannian metric) on
F (�(F2) [X) a standard left (right) parabolic transposition.
Remark 4.5. Notice that we cannot use Lemma 3:1 directly to the foliation
�(F2) since it is not a foliation by disks.
4.5. Parabolic Turbulization along K on a Three-Sphere S3
Lemma 4.6. For any topological type of the knot K there is a parabolic folia-
tion FK on S3 such that FK is a parabolic turbulization along K of the parabolic
Reeb foliation on S3 (see Proposition 4:1).
P r o o f. Consider a standard presentation of K where n is the number of
strings in it. Recall that K is a subset of the union of vertical circles fxi � S1g
188 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
D
2
j
xj
xj+1
D (1/3)
2
Fig. 9: A disk D2(1
3
)� f0g:
everywhere except some neighborhoods of the inverse images of double points of
K in the frontal projection.
In the frontal projection, let t1; : : : ; tN denote a set of t-coordinates of double
points of K.
Write the presentation of K as a product K = ��11 ��12 : : : ��1
N
. Recall that
with each vertex of the regular polygon we associated the disk with the center
at the vertex and with radius "(see p. 15). For each �j consider a disk D2
j
with
the center at the median of edge xj�1xj and radius d(xj�1xj)=2 + 2". We choose
a disk with this radius for the small disks with the centers at vertices and of radius
" to be inside it (see Fig. 9).
Notice that since xj are the vertices of regular polygon, the points xj�1 and
xj are the only points from the set fx1; x2; : : : ; xng, which are inside the disk D2
j
.
Consider a set of disjoint intervals I(tj); j = 1 : : : N � [0; 1] such that tj 2 I(tj).
On S3 de�ne a standard turbulization with n strings and a metric (see
Prop. 4:2) such that Fn is parabolic with respect to it. For each j let (D2(1
3
) �
[0; 1];F 0; g) be a left (or right) standard parabolic transposition depending on
a degree of corresponding �j . Denote the radius of D2
j
by rj and the length of
the segment I(tj) by dj . Consider the map
Fj : D
2(
1
3
)� [0; 1]! D2
j
� I(tj)
which is given by the formula
Fj((r; �); t) = ((
3r
rj
; �);
t
dj
):
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 189
V. Krouglov
This map de�nes a foliation Fj(F
0) inside the ball D2
j
� I(tj). An inverse map
F�1
j
de�nes on D2
j
� I(tj) such a metric that Fj(F
0) is parabolic with respect to
it. Since in the neighborhood of gluing the foliation is a direct product foliation
and the metric is a direct product metric, it glues correctly to Fn. Denote the
obtained foliation by FK . This foliation is parabolic with respect to the glued
metric.
5. Gluing the Solid Torus
P r o o f of Theorem 1:1. In order to make the proof of the theorem complete
we have to perform a Dehn surgery on a knot K.
Consider the foliation FK on S3. Let N denote such a tubular neighborhood
of K that @N = T 2 is a leaf of FK . Let X = S3nN and consider an arbitrary
di�eomorphism f
f@X ! @(D2 � S1):
This di�eomorphism is de�ned up to isotopy by the map it induces in �rst
homology
f� : H1(T
2)! H1(T
2) f� 2 SL2(Z):
In particular, we may think that f =
�
a b
c d
�
is a linear map.
OnD2�S1 de�ne a parabolic foliation and a metric as in Lemma 3:3. The met-
ric is euclidian in some neighborhood of boundary torus. Therefore f de�nes the
following metric on @X:
G =
�
a2 + c2 ac+ bd
ac+ bd b2 + d2
�
:
In its turn, there is a metric H =
�
1 0
0 1
�
on @X. To interpolate from G to H,
consider the union X[T 2� [0; 1][fD
2�S1. By Corollary 3:2 there is a metric on
T 2� [0; 1] which deforms G to H in such a way that the foliation by tori T 2�fptg
is parabolic. Since in the (one-sided) neighborhoods of @X and @(D2 � S1) the
metrics are direct product metrics, on the unionX[T 2�[0; 1][fD
2�S1 we obtain
a smooth Riemannian metric. As X [ T 2 � [0; 1] [f D
2 � S1 is di�eomorphic to
X [fD
2�S1, every closed oriantable three-manifold admits a parabolic foliation.
190 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2
Parabolic Foliations on Three-Manifolds
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