Conformal geometry of foliations (slajdy wykładu w Tambara

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CONFORMAL GEOMETRY of FOLIATIONS
Paweł Walczak
Uniwersytet Łódzki
Japan, October 2009
Paweł Walczak Uniwersytet Łódzki
General problem
HERE:
M – a (compact) oriented Riemannian manifold, n = dim M
F – an oriented foliation on M, dim F = n − 1
(P) – a geometric property of submanifolds, e.g.
totally geodesic, totally umbilical etc.
Definition
F is a (P)-foliation whenever all the leaves of F have property
(P).
Problem
Given M and (P), does there exist (P)-foliations on M ?
Known results
Theorem
1. (Zeghib) KM = const < 0, (P) = totally geodesic, then no
(P)-foliations on M.
Theorem
2. (obvious) KM = const > 0, (P) = totally umbilical (⊃ ”totally
geodesic”, then no (P)-foliations on M.
Theorem
3. (R. Langevin, P.W.) If KM < 0 (even, RicM < 0), (P) =
totally umbilical, then no (P)-foliations on M.
Proof of Theorem 3.
Follows directly from the well known integral formula
Z
M
(2σ2 − RicM (N))d vol = 0,
where
P
σ2 = i<j ki kj and ki - principal curvatures of the leaves
and N - a unit normal vector of F.
Particular problem
Remark
The property (P) = totally umbilical is conformally invariant.
Problem
Weaken the property (P) in conformal geometry to get
existence results. Classify (P)-foliations whenever exist.
FROM NOW: KM = const, dim M = 3
Local conformal invariants (LCI) for surfaces - 1
After: Cairns, Sharp and Webb, 1994
Tresse,1892
Fialkow, 1944
Bryant, 1984
k1 , k2 - principal curvatures (k1 > k2 )
µ := (k1 − k2 )/2
Xi = unit principal vector corresponding to ki
ξi = Xi /µ
θi = Xi (ki )/µ2 - conformal principal curvatures
(ω1 , ω2 ) - 1-forms dual to (ξ1 , ξ2 )
LCI - 2
Ψ=A+B
is called here the Bryant invariant,
where:
A = (∆H + 2µ2 H)/µ3
(H = mean curvature),
is the first variation for the Willmore functional:
W (S) =
R
S (k1
− k2 )2 d vol
and
B = (1/2)(θ12 − θ22 + ξ1 (θ1 ) + ξ2 (θ2 ))
LCI-3
Define (5 × 5) matrices A1 and A2 by


θ1 /2 −(1 + Ψ)/2 b/2 θ1 /2
0
 1
0
0
−1 (1 + Ψ)/2




0
0
0
−b/2 
A1 =  0


 0
1
0
0
−θ1 /2 
0
−1
0
0
−θ1 /2
(1)
and


−θ2 /2 −c/2 −(1 − Ψ)/2 θ2 /2
0
 0

0
0
0
c/2




0
0
1
(1 − Ψ)/2 ,
A2 =  1


 0
0
−1
0
−θ2 /2 
0
0
−1
0
θ2 /2
where b = −θ1 θ2 + ξ2 (θ1 ) and c = θ1 θ2 + ξ1 (θ2 ).
(2)
LCI-4
Theorem
4. (Fialkow) If, on a simply connected domain U ⊂ R2 , the form
ω = A1 ω1 + A2 ω2
(3)
satisfies the structural equation
dω + (1/2)[ω, ω] = 0,
(4)
then there exists an immersion ι : U → R3 for which S = ι(U)
realizes these forms and functions as local conformal invariants. S
is unique up to a Möbius transformation.
Dupin cyclides
Definition
A surface is called a Dupin cyclid whenever θ1 = θ2 = 0 (then:
Ψ = const.).
Dupin cyclides:
(1) are conformal images of tori, cylinders and cones of revolution,
(2) enjoy the spherical two piece property = STPP and can be
characterized by it (Banchoff, 1970)
Dupin foliations
Theorem
5. (R. Langevin, P.W., 2008) If KM = const. 6= 0, then there are
no Dupin foliations on M.
Proof. (1) If KM > 0, the theorem follows directly from STPP +
the Novikov Theorem.
(2) If KM < 0, the result can be obtained by lifting a foliation to
the universal cover M̃, the fact that the action of Γ = the group of
covering transformations on the boundary M̃(∞) is minimal and
the fact that some Dupin cyclides have to intersect M̃(∞)
transversely.
CCI-foliations
Definition
A surface is said to be of constant conformal invariants (=
CCI) if the functions θ1 , θ2 and Ψ are constant.
Theorem
6. (A. Bartoszek, P.W., 2008)
(i) If θ1 and θ2 are constant, then θ1 θ2 = 0.
(ii) If a surface is CCI, and θ12 + θ22 > 0, then Ψ = ±2.
(iii) If a surface is CCI, then it is either (a piece of a conformal
image of) a Dupin cyclid or a cylinder over a logarithmic spiral (=
loxodromic cylinder).
(iv) There are no CCI foliations of M with KM = const. 6= 0.
Proof of Theorem 6.
Proof. (1) + (2) follows from calculations and the structural
equation (4).
(3) follows from the calculation of local conformal invariants for
loxodromic cylinders and uniquness part of Fialkow Theorem.
(4) is obtained in the same way as the Langevin-W result on Dupin
foliations.
Canal surfaces
Definition
Canal surfaces are envelopes of 1-parameter families of spheres.
They can be characterized by the condition
θ1 · θ2 = 0.
and are ”foliated” by characteristic circles, the lines of curvature
corresponding to the vanishing identically conformal principal
curvature (say, θ1 = 0).
Special canals
Definition
A canal surface is said to be special iff the other conformal
principal curvature (say, θ2 ) is constant along characteristic
circles.
Special canals are classified and split into three classes, conformal
images of
(1) surfaces of revolution (|Ψ| < 2),
(2) cylinders over place curves (|Ψ| = 2),
(3) cones over plane curves (|Ψ| > 2).
Canal Reeb foliations
The standard Reeb foliation can be made of (conformal images of)
surfaces of revolution, therefore of special canal surfaces. therefore
of canal surfaces. So,
Proposition
there exist canal foliations (even, special canal foliations) of S 3 .
Canal foliations of S 3
Theorem
7. (R. Langevin, P. W.) Any canal foliation of S 3 is obtained
from a standard Reeb foliation by inserting a zone Z ' T 2 × [0, 1],
a union of toral and cylindrical leaves.
Theorem
8. (R. L., P. W.) At least one leaf of a canal foliation is a Dupin
cyclide.
Essential zones
The maximal zone Zmax may contain essential zones
Zi ' T 2 × [0, 1] containing cylindrical leaves which give to the two
limit tori the opposite orientations:
Spiralling components
The zone Zmax may contain also spiralling components, where the
cylindrical leaves induce on boundary tori the same orientation::
The maximal zone
The following completes the classification of (special) canal
foliations of S 3 :
Theorem
9. (R. L., P. W.) The zone Zmax contains a maximal product
lamination T 2 × K , K a compact subset of the interval; the toral
leaves are nested, that is all the toral leaves are unknotted, and
each toral leaf is contained in one solid torus bounded by the other
toral leaves. The complement of this lamination is a union of:
- a finite number of essential zones,
- a finite or countable number of spiralling components.
Remaining problem
Problem
Do they exist canal foliations on closed hyperbolic manifolds ?
If ”YES”, classify them.
If ”NOT”, what is the consecutive reduction of the condition
(P) providing the existence ?
Bibliography 1
R. Bryant, A duality theorem for Willmore surfaces, J. Diff.
Geom. 20 (1084), 23 – 53.
G. Cairns, R. Sharpe, L. Webb, Conformal invariants for curves
and surfaces in three dimensional space forms, Rocky
Mountain J. Math. 24 (1994), 933 – 959.
A. Fialkov, Conformal differential geometry of a subspace,
Trans. Amer. Math. Soc. 56 (1944), 309 – 433.
A. Tresse, Sur les invariants différentiels d’une surface par
rapport aux transformations conformes de l’espace, C.R. Acad.
Sci. Paris 114 (1892), 948 – 950.
Bibliography 2
A. Bartoszek, P.W., Foliations by surfaces of a peculiar class,
Ann. Polon. Math., 94 (2008), 89 – 95.
A. Bartoszek, R. Langevin, P. W., Special canal surfaces of S 3 ,
in preparation.
R. Langevin, P. W., Conformal geometry of foliations, Geom.
Dedicata, 132 (2008), 135–178.
R. Langevin, P. W., Canal foliations of S 3 , Preprint, 2009.

Conformal geometry of foliations (slajdy wykładu w Tambara

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