Katrin FAESSLER

A stretch map on the Heisenberg group
Katrin Fässler
joint work with
Zoltán Balogh and Ioannis Platis
HCAA 2012
March 5–9, 2012
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 1 / 23

Radial stretch maps
On the complex plane
fk(z) = |z| k−1 z, (0 < k < 1).
a prototype for quasiconformal mappings
extremal for various problems.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 2 / 23

Radial stretch maps
On the complex plane
fk(z) = |z| k−1 z, (0 < k < 1).
a prototype for quasiconformal mappings
extremal for various problems.
Question
Radial stretch map on the sub-Riemannian Heisenberg group?
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 2 / 23

The setting: Heisenberg group
Definition
The Heisenberg group H 1 can be modeled on C × R such that the group
law reads
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 3 / 23

The setting: Heisenberg group
Definition
The Heisenberg group H 1 can be modeled on C × R such that the group
law reads
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.
H 1 can be endowed with a left invariant metric
where pH = (|z| 4 + t 2 ) 1
4 .
dH(p, q) := q −1 ∗ pH, p, q ∈ H 1 ,
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 3 / 23

The setting: Heisenberg group
Definition
The Heisenberg group H 1 can be modeled on C × R such that the group
law reads
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.
H 1 can be endowed with a left invariant metric
where pH = (|z| 4 + t 2 ) 1
4 .
For fixed s > 0, one has
where δs(z, t) = (sz, s 2 t).
dH(p, q) := q −1 ∗ pH, p, q ∈ H 1 ,
dH(δs(p), δs(q)) = s dH(p, q), p, q ∈ H 1 ,
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 3 / 23

Objects of study: quasiconformal maps on (H 1 , dH)
Definition (metric definition)
A homeomorphism f : Ω → Ω ′ between domains in H 1 is called
quasiconformal (QC) if there is 1 ≤ H < ∞ such that
H(p, f ) := lim sup
r→0+
Katrin Fässler (University of Helsinki) A stretch map on H 1
supdH(p,q)=r dH(f (p), f (q))
< H, for all p ∈ Ω.
infdH(p,q)=r dH(f (p), f (q))
HCAA 2012 4 / 23

Objects of study: quasiconformal maps on (H 1 , dH)
Definition (metric definition)
A homeomorphism f : Ω → Ω ′ between domains in H 1 is called
quasiconformal (QC) if there is 1 ≤ H < ∞ such that
H(p, f ) := lim sup
r→0+
supdH(p,q)=r dH(f (p), f (q))
< H, for all p ∈ Ω.
infdH(p,q)=r dH(f (p), f (q))
Motivated by Mostow’s work on rigidity of locally symmetric spaces.
Rich theory developed by Korányi, Pansu and Reimann.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 4 / 23

Horizontal structure on the Heisenberg group
A contact structure on H 1 is given by
τ = −izdz + izdz + dt.
The horizontal bundle HH 1 is the subbundle of the tangent bundle T H 1
with fibers
HpH 1 = kerτp.
A complex basis for the complexified space H C p H 1 is given by
Z = ∂
∂z
∂ ∂
+ iz , Z =
∂t ∂z
Katrin Fässler (University of Helsinki) A stretch map on H 1
− iz ∂
∂t .
HCAA 2012 5 / 23

Horizontal structure on the Heisenberg group
A contact structure on H 1 is given by
τ = −izdz + izdz + dt.
The horizontal bundle HH 1 is the subbundle of the tangent bundle T H 1
with fibers
HpH 1 = kerτp.
A complex basis for the complexified space H C p H 1 is given by
Z = ∂
∂z
∂ ∂
+ iz , Z =
∂t ∂z
− iz ∂
∂t .
Objects which behave well with respect to this structure:
Horizontal curves: γ : [a, b] → H 1 with ˙γ(s) ∈ H γ(s)H 1 a.e.
Contact transformations: f : H 1 ⊇ Ω → Ω ′ ⊆ H 1 , with f ∗ τ = λτ.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 5 / 23

Contact property of QC maps
Contact condition (f∗,pHpH 1 ⊆ H f (p)H 1 ) for a map f = (fI , f3) reads
f I ZfI − fI Zf I = −iZf3.
Quasiconformal maps on H 1 are contact.
First proved by Korányi and Reimann for differentiable QC maps. It holds
for arbitrary QC maps in a weak sense.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 6 / 23

Contact property of QC maps
Contact condition (f∗,pHpH 1 ⊆ H f (p)H 1 ) for a map f = (fI , f3) reads
f I ZfI − fI Zf I = −iZf3.
Quasiconformal maps on H 1 are contact.
First proved by Korányi and Reimann for differentiable QC maps. It holds
for arbitrary QC maps in a weak sense.
Contact property of QC maps on H 1 . . .
extra information on the form of such maps,
obstruction for the construction of such maps.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 6 / 23

Definition of a Heisenberg stretch
First try:
gk(p) := δ k−1
p (p)
H
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 7 / 23

Definition of a Heisenberg stretch
First try:
gk(p)H = p k H ,
gk(p) := δ k−1
p (p)
H
BUT gk is not contact, hence not QC.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 7 / 23

Definition of a Heisenberg stretch
First try:
gk(p)H = p k H ,
gk(p) := δ k−1
p (p)
H
BUT gk is not contact, hence not QC.
More general
does not work either.
hk(p) := δ ϕk(p)(p)
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 7 / 23

Definition of a Heisenberg stretch II
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 8 / 23

Definition of a Heisenberg stretch II
The planar stretch fk(z) = |z| k−1 z is
given in logarithmic coordinates by
(ln |z|, arg(z)) ↦→ (k ln |z|, arg(z)).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 8 / 23

Definition of a Heisenberg stretch II
The planar stretch fk(z) = |z| k−1 z is
given in logarithmic coordinates by
(ln |z|, arg(z)) ↦→ (k ln |z|, arg(z)).
Second try:
Use “logarithmic coordinates” on the Heisenberg group!
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 8 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
◮ ξ = 0: Korányi unit sphere
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
◮ ξ = 0: Korányi unit sphere
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1
3 (π − θ − 2φ).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
◮ ξ = 0: Korányi unit sphere
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1
3 (π − θ − 2φ).
Particular horizontal curves:
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
◮ ξ = 0: Korányi unit sphere
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1
3 (π − θ − 2φ).
Particular horizontal curves:
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (ξ, s, η);
Katrin Fässler (University of Helsinki) A stretch map on H 1
Picture
HCAA 2012 9 / 23

“Logarithmic coordinates”
(z, t) =
where (ξ, ψ, η) ∈ R × (− π π
2 , 2 ) × R.
i cos ψe ξ+i(ψ−3η)
2 , − sin ψe ξ
,
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)
◮ η = 0: boundary of “standard flat pack”
◮ ψ = 0: plane C × {0}
◮ ξ = 0: Korányi unit sphere
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1
3 (π − θ − 2φ).
Particular horizontal curves:
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (ξ, s, η);
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (s, ψ, η −
Katrin Fässler (University of Helsinki) A stretch map on H 1
Picture
tan ψ
3 s) (“radials”). Picture
HCAA 2012 9 / 23

Figure: “Radials” connecting two Korányi spheres, each curve belongs to the
Legendrian foliation of a Heisenberg cone associated to the parameter ψ.

Contact transformations in logarithmic coordinates
Contact form
We write mappings on H 1 as
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 11 / 23

Contact transformations in logarithmic coordinates
Contact form
We write mappings on H 1 as
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))
To define a stretch map fk, 0 < k < 1, set
Ξ(ξ, ψ, η) = kξ.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 11 / 23

Contact transformations in logarithmic coordinates
Contact form
We write mappings on H 1 as
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))
To define a stretch map fk, 0 < k < 1, set
Ξ(ξ, ψ, η) = kξ.
Then the contact condition yields
H(ξ, ψ, η) = H(ξ, η), Ψ(ξ, ψ, η) = tan −1
Hη(ξ, η) tan ψ − 3Hξ(ξ, η)
.
k
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 11 / 23

Contact transformations in logarithmic coordinates
Contact form
We write mappings on H 1 as
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))
To define a stretch map fk, 0 < k < 1, set
Ξ(ξ, ψ, η) = kξ.
Then the contact condition yields
H(ξ, ψ, η) = H(ξ, η), Ψ(ξ, ψ, η) = tan −1
Hη(ξ, η) tan ψ − 3Hξ(ξ, η)
.
k
The stretch map is obtained by setting
H(ξ, η) = η.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 11 / 23

Heisenberg stretch map
Definition
Let 0 < k < 1. The Heisenberg stretch map is given by
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ
( k ), η).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 12 / 23

Heisenberg stretch map
Definition
Let 0 < k < 1. The Heisenberg stretch map is given by
Properties:
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ
( k ), η).
fk is quasiconformal with fk(p)H = p k H ,
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 12 / 23

Heisenberg stretch map
Definition
Let 0 < k < 1. The Heisenberg stretch map is given by
Properties:
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ
( k ), η).
fk is quasiconformal with fk(p)H = p k H ,
fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0),
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 12 / 23

Heisenberg stretch map
Definition
Let 0 < k < 1. The Heisenberg stretch map is given by
Properties:
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ
( k ), η).
fk is quasiconformal with fk(p)H = p k H ,
fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0),
z
f−1(z, t) = |z| 2 −t , −it |z| 4 +t2
is 1-QC inversion in the Korányi unit
sphere with inversion relations
dH(f−1(p), f−1(q)) = dH(p, q)
pHqH
Katrin Fässler (University of Helsinki) A stretch map on H 1
and f−1(p)H = 1
.
pH
HCAA 2012 12 / 23

Analytic characterization of QC maps on H 1
Theorem (Korányi, Reimann, Pansu, Vodop’yanov)
A homeomorphism f : Ω → Ω ′ between domains in H 1 is quasiconformal if
and only if
f ∈ HW 1,4
loc (Ω),
f is a weakly contact map,
Here,
∃ 1 ≤ K < ∞ such that DHf (p) 4 ≤ KJ(p, f ) a.e.
DHf 4 = (|ZfI | + |¯ZfI |) 4
Katrin Fässler (University of Helsinki) A stretch map on H 1
and J(·, f ) = (|ZfI | 2 − |¯ZfI | 2 ) 2 .
HCAA 2012 13 / 23

Analytic characterization of QC maps on H 1
Theorem (Korányi, Reimann, Pansu, Vodop’yanov)
A homeomorphism f : Ω → Ω ′ between domains in H 1 is quasiconformal if
and only if
f ∈ HW 1,4
loc (Ω),
f is a weakly contact map,
Here,
∃ 1 ≤ K < ∞ such that DHf (p) 4 ≤ KJ(p, f ) a.e.
DHf 4 = (|ZfI | + |¯ZfI |) 4
and J(·, f ) = (|ZfI | 2 − |¯ZfI | 2 ) 2 .
Definition (Distortion)
|ZfI
K(·, f ) =
| + |ZfI |
|ZfI | − |ZfI |
and Kf = K(·, f )∞.
Note: H(·, f ) = K(·, f ) a.e.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 13 / 23

Proposition
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion
K((ξ, ψ, η), fk) = 1 + tan2 ψ
k 2 + tan 2 ψ
Katrin Fässler (University of Helsinki) A stretch map on H 1
1
and Kfk = .
k2 HCAA 2012 14 / 23

Proposition
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion
K((ξ, ψ, η), fk) = 1 + tan2 ψ
k 2 + tan 2 ψ
Let F be the class of QC maps
1
and Kfk = .
k2 g : A(a, b) := {p ∈ H 1 : a < pH < b} → A(a k , b k )
which map the inner and outer boundary components of A(a, b)
homeomorphically to the respective boundary components of A(a k , b k ).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 14 / 23

Proposition
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion
K((ξ, ψ, η), fk) = 1 + tan2 ψ
k 2 + tan 2 ψ
Let F be the class of QC maps
1
and Kfk = .
k2 g : A(a, b) := {p ∈ H 1 : a < pH < b} → A(a k , b k )
which map the inner and outer boundary components of A(a, b)
homeomorphically to the respective boundary components of A(a k , b k ).
Question
Is fk extremal (i.e., a minimizer for maximal or mean distortion) in F?
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 14 / 23

Extremality of the stretch I (mean distortion)
Theorem
For all g ∈ F we have that
K(p, fk)
A(a,b)
2 ρ0(p) 4
dµ(p) ≤
with ρ0 = (log b
Here
Ω hρ4 0
a )−1 ∇0 log( · H)0.
dµ =
Proof by modulus method.
K(p, g)
A(a,b)
2 ρ0(p) 4 dµ(p)
Ω hρ4 0 dµ
Ω ρ4 0 dµ and µ denotes the Haar measure on H1 .
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 15 / 23

Modulus of curve families – a ‘quasi-invariant’ for QC maps
Definition
The 4-modulus of a curve family Γ on H1 is given by
M4(Γ) = inf ρ
ρ∈adm(Γ)
4 dµ,
adm(Γ) = {ρ : H 1 → [0, ∞] Borel with
For γ = (z, t) horizontal:
γ ρ dℓ = b
a ρ(γ(s))|˙z(s)| ds.
Katrin Fässler (University of Helsinki) A stretch map on H 1
γ
H 1
ρ dℓ ≥ 1 for γ ∈ Γ loc. rectif.}.
HCAA 2012 16 / 23

Modulus of curve families – a ‘quasi-invariant’ for QC maps
Definition
The 4-modulus of a curve family Γ on H1 is given by
M4(Γ) = inf ρ
ρ∈adm(Γ)
4 dµ,
adm(Γ) = {ρ : H 1 → [0, ∞] Borel with
For γ = (z, t) horizontal:
γ ρ dℓ = b
a ρ(γ(s))|˙z(s)| ds.
Cf. work by Markina and Vodop’yanov, and by Tang.
Theorem
γ
H 1
ρ dℓ ≥ 1 for γ ∈ Γ loc. rectif.}.
Let f : Ω → Ω ′ be a QC map between domains in H1 and let Γ be a family
of curves in Ω. Then
M4(f (Γ)) ≤ K((z, t), f ) 2 ρ 4 (z, t) dµ(z, t) for all ρ ∈ adm(Γ).
Ω
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 16 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Katrin Fässler (University of Helsinki) A stretch map on H 1
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
HCAA 2012 17 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 17 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 17 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2 ρ 4 0 dµ.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 17 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ.
Thus,
K(·, fk) 2 ρ 4
0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 17 / 23

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and
thus
f ◦γ ρ′ dℓ ≥
γ ρ dℓ with ρ′ ρ
=
||ZfI |−|¯ZfI || ◦ f −1 .
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and
thus
f ◦γ ρ′ dℓ ≥
γ ρ dℓ with ρ′ ρ
=
||ZfI |−|¯ZfI || ◦ f −1 .
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ
◮ K(·, fk) is constant along each γψ,η ∈ Γ0;

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and
thus
f ◦γ ρ′ dℓ ≥
γ ρ dℓ with ρ′ ρ
=
||ZfI |−|¯ZfI || ◦ f −1 .
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ
◮ K(·, fk) is constant along each γψ,η ∈ Γ0;
◮ integration formula:
2 ln b
h dµ =
h(γψ,η(s))|˙zψ,η(s)| 4
ds 12 cos 2 (ψ)dψdη.
A(a,b)
Q
2 ln a

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Katrin Fässler (University of Helsinki) A stretch map on H 1
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
HCAA 2012 19 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 19 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 19 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2 ρ 4 0 dµ.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 19 / 23

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ.
Thus,
K(·, fk) 2 ρ 4
0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 19 / 23

Extremality of the stretch II (maximal distortion)
M4(Γ) = π 2 ln
b −3
a
(cf. Korányi, Reimann) and thus
and M4(fk(Γ)) = k −3 π 2 ln
b −3
a ,
M4(fk(Γ)) K 2
fk M4(Γ).
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 20 / 23

Extremality of the stretch II (maximal distortion)
M4(Γ) = π 2 ln
b −3
a
(cf. Korányi, Reimann) and thus
and M4(fk(Γ)) = k −3 π 2 ln
b −3
a ,
M4(fk(Γ)) K 2
fk M4(Γ).
Question
Is the stretch map fk extremal for the maximal distortion within F?
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 20 / 23

Extremality of the stretch II (maximal distortion)
M4(Γ) = π 2 ln
b −3
a
(cf. Korányi, Reimann) and thus
and M4(fk(Γ)) = k −3 π 2 ln
b −3
a ,
M4(fk(Γ)) K 2
fk M4(Γ).
Question
Is the stretch map fk extremal for the maximal distortion within F?
Proposition
The stretch map fk is extremal for the maximal distortion in the class F0
of C 1 orientation- and sphere-preserving maps g ∈ F which map the t-axis
to the t-axis, i.e., Kfk ≤ Kg for all g ∈ F0.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 20 / 23

Extremality of the stretch II (maximal distortion)
M4(Γ) = π 2 ln
b −3
a
(cf. Korányi, Reimann) and thus
and M4(fk(Γ)) = k −3 π 2 ln
b −3
a ,
M4(fk(Γ)) K 2
fk M4(Γ).
Question
Is the stretch map fk extremal for the maximal distortion within F?
Proposition
The stretch map fk is extremal for the maximal distortion in the class F0
of C 1 orientation- and sphere-preserving maps g ∈ F which map the t-axis
to the t-axis, i.e., Kfk ≤ Kg for all g ∈ F0.
Proof idea: write contact transformation g ∈ F0 in logarithmic
coordinates. Sphere-preserving: Ξ = Ξ(ξ), apply MVT: ∃ξ0 with
Ξξ(ξ0) = k. Formula for distortion. Find p0 with K(p0, g) ≥ Kfk .
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 20 / 23

Thank you for your attention!
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 21 / 23

References
A. Korányi and H. M. Reimann
Horizontal normal vectors and conformal capacity of spherical rings in
the Heisenberg group
Bull. Sci. Math. (2), 111(1):3–21, 1987
A. Korányi and H. M. Reimann
Foundations for the theory of quasiconformal mappings on the
Heisenberg group
Adv. Math, 111(1):1–87, 1995
I. D. Platis
The geometry of complex hyperbolic packs
Math. Proc. Cambridge Philos. Soc., 147(1):205–234, 2009
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 22 / 23

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Figure: Curves in the Legendrian foliation of a Korányi sphere