Katrin FAESSLER
Katrin FAESSLER
Katrin FAESSLER
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A stretch map on the Heisenberg group<br />
<strong>Katrin</strong> Fässler<br />
joint work with<br />
Zoltán Balogh and Ioannis Platis<br />
HCAA 2012<br />
March 5–9, 2012<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 1 / 23
Radial stretch maps<br />
On the complex plane<br />
fk(z) = |z| k−1 z, (0 < k < 1).<br />
a prototype for quasiconformal mappings<br />
extremal for various problems.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 2 / 23
Radial stretch maps<br />
On the complex plane<br />
fk(z) = |z| k−1 z, (0 < k < 1).<br />
a prototype for quasiconformal mappings<br />
extremal for various problems.<br />
Question<br />
Radial stretch map on the sub-Riemannian Heisenberg group?<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 2 / 23
The setting: Heisenberg group<br />
Definition<br />
The Heisenberg group H 1 can be modeled on C × R such that the group<br />
law reads<br />
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),<br />
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 3 / 23
The setting: Heisenberg group<br />
Definition<br />
The Heisenberg group H 1 can be modeled on C × R such that the group<br />
law reads<br />
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),<br />
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.<br />
H 1 can be endowed with a left invariant metric<br />
where pH = (|z| 4 + t 2 ) 1<br />
4 .<br />
dH(p, q) := q −1 ∗ pH, p, q ∈ H 1 ,<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 3 / 23
The setting: Heisenberg group<br />
Definition<br />
The Heisenberg group H 1 can be modeled on C × R such that the group<br />
law reads<br />
p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )),<br />
for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R.<br />
H 1 can be endowed with a left invariant metric<br />
where pH = (|z| 4 + t 2 ) 1<br />
4 .<br />
For fixed s > 0, one has<br />
where δs(z, t) = (sz, s 2 t).<br />
dH(p, q) := q −1 ∗ pH, p, q ∈ H 1 ,<br />
dH(δs(p), δs(q)) = s dH(p, q), p, q ∈ H 1 ,<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 3 / 23
Objects of study: quasiconformal maps on (H 1 , dH)<br />
Definition (metric definition)<br />
A homeomorphism f : Ω → Ω ′ between domains in H 1 is called<br />
quasiconformal (QC) if there is 1 ≤ H < ∞ such that<br />
H(p, f ) := lim sup<br />
r→0+<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
supdH(p,q)=r dH(f (p), f (q))<br />
< H, for all p ∈ Ω.<br />
infdH(p,q)=r dH(f (p), f (q))<br />
HCAA 2012 4 / 23
Objects of study: quasiconformal maps on (H 1 , dH)<br />
Definition (metric definition)<br />
A homeomorphism f : Ω → Ω ′ between domains in H 1 is called<br />
quasiconformal (QC) if there is 1 ≤ H < ∞ such that<br />
H(p, f ) := lim sup<br />
r→0+<br />
supdH(p,q)=r dH(f (p), f (q))<br />
< H, for all p ∈ Ω.<br />
infdH(p,q)=r dH(f (p), f (q))<br />
Motivated by Mostow’s work on rigidity of locally symmetric spaces.<br />
Rich theory developed by Korányi, Pansu and Reimann.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 4 / 23
Horizontal structure on the Heisenberg group<br />
A contact structure on H 1 is given by<br />
τ = −izdz + izdz + dt.<br />
The horizontal bundle HH 1 is the subbundle of the tangent bundle T H 1<br />
with fibers<br />
HpH 1 = kerτp.<br />
A complex basis for the complexified space H C p H 1 is given by<br />
Z = ∂<br />
∂z<br />
∂ ∂<br />
+ iz , Z =<br />
∂t ∂z<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
− iz ∂<br />
∂t .<br />
HCAA 2012 5 / 23
Horizontal structure on the Heisenberg group<br />
A contact structure on H 1 is given by<br />
τ = −izdz + izdz + dt.<br />
The horizontal bundle HH 1 is the subbundle of the tangent bundle T H 1<br />
with fibers<br />
HpH 1 = kerτp.<br />
A complex basis for the complexified space H C p H 1 is given by<br />
Z = ∂<br />
∂z<br />
∂ ∂<br />
+ iz , Z =<br />
∂t ∂z<br />
− iz ∂<br />
∂t .<br />
Objects which behave well with respect to this structure:<br />
Horizontal curves: γ : [a, b] → H 1 with ˙γ(s) ∈ H γ(s)H 1 a.e.<br />
Contact transformations: f : H 1 ⊇ Ω → Ω ′ ⊆ H 1 , with f ∗ τ = λτ.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 5 / 23
Contact property of QC maps<br />
Contact condition (f∗,pHpH 1 ⊆ H f (p)H 1 ) for a map f = (fI , f3) reads<br />
f I ZfI − fI Zf I = −iZf3.<br />
Quasiconformal maps on H 1 are contact.<br />
First proved by Korányi and Reimann for differentiable QC maps. It holds<br />
for arbitrary QC maps in a weak sense.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 6 / 23
Contact property of QC maps<br />
Contact condition (f∗,pHpH 1 ⊆ H f (p)H 1 ) for a map f = (fI , f3) reads<br />
f I ZfI − fI Zf I = −iZf3.<br />
Quasiconformal maps on H 1 are contact.<br />
First proved by Korányi and Reimann for differentiable QC maps. It holds<br />
for arbitrary QC maps in a weak sense.<br />
Contact property of QC maps on H 1 . . .<br />
extra information on the form of such maps,<br />
obstruction for the construction of such maps.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 6 / 23
Definition of a Heisenberg stretch<br />
First try:<br />
gk(p) := δ k−1<br />
p (p)<br />
H<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 7 / 23
Definition of a Heisenberg stretch<br />
First try:<br />
gk(p)H = p k H ,<br />
gk(p) := δ k−1<br />
p (p)<br />
H<br />
BUT gk is not contact, hence not QC.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 7 / 23
Definition of a Heisenberg stretch<br />
First try:<br />
gk(p)H = p k H ,<br />
gk(p) := δ k−1<br />
p (p)<br />
H<br />
BUT gk is not contact, hence not QC.<br />
More general<br />
does not work either.<br />
hk(p) := δ ϕk(p)(p)<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 7 / 23
Definition of a Heisenberg stretch II<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 8 / 23
Definition of a Heisenberg stretch II<br />
The planar stretch fk(z) = |z| k−1 z is<br />
given in logarithmic coordinates by<br />
(ln |z|, arg(z)) ↦→ (k ln |z|, arg(z)).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 8 / 23
Definition of a Heisenberg stretch II<br />
The planar stretch fk(z) = |z| k−1 z is<br />
given in logarithmic coordinates by<br />
(ln |z|, arg(z)) ↦→ (k ln |z|, arg(z)).<br />
Second try:<br />
Use “logarithmic coordinates” on the Heisenberg group!<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 8 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
◮ ξ = 0: Korányi unit sphere<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
◮ ξ = 0: Korányi unit sphere<br />
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)<br />
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1<br />
3 (π − θ − 2φ).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
◮ ξ = 0: Korányi unit sphere<br />
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)<br />
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1<br />
3 (π − θ − 2φ).<br />
Particular horizontal curves:<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
◮ ξ = 0: Korányi unit sphere<br />
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)<br />
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1<br />
3 (π − θ − 2φ).<br />
Particular horizontal curves:<br />
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (ξ, s, η);<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
Picture<br />
HCAA 2012 9 / 23
“Logarithmic coordinates”<br />
(z, t) =<br />
where (ξ, ψ, η) ∈ R × (− π π<br />
2 , 2 ) × R.<br />
<br />
i cos ψe ξ+i(ψ−3η)<br />
2 , − sin ψe ξ<br />
,<br />
Simple description of Korányi spheres ξ = 2 ln ((z, t)H).<br />
Derived from coordinates on H 2 C which extend to ∂H2 C (Platis)<br />
◮ η = 0: boundary of “standard flat pack”<br />
◮ ψ = 0: plane C × {0}<br />
◮ ξ = 0: Korányi unit sphere<br />
Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ)<br />
(Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1<br />
3 (π − θ − 2φ).<br />
Particular horizontal curves:<br />
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (ξ, s, η);<br />
◮ s ↦→ (ξ(s), ψ(s), η(s)) = (s, ψ, η −<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
Picture<br />
tan ψ<br />
3 s) (“radials”). Picture<br />
HCAA 2012 9 / 23
Figure: “Radials” connecting two Korányi spheres, each curve belongs to the<br />
Legendrian foliation of a Heisenberg cone associated to the parameter ψ.
Contact transformations in logarithmic coordinates<br />
Contact form<br />
We write mappings on H 1 as<br />
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).<br />
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 11 / 23
Contact transformations in logarithmic coordinates<br />
Contact form<br />
We write mappings on H 1 as<br />
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).<br />
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))<br />
To define a stretch map fk, 0 < k < 1, set<br />
Ξ(ξ, ψ, η) = kξ.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 11 / 23
Contact transformations in logarithmic coordinates<br />
Contact form<br />
We write mappings on H 1 as<br />
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).<br />
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))<br />
To define a stretch map fk, 0 < k < 1, set<br />
Ξ(ξ, ψ, η) = kξ.<br />
Then the contact condition yields<br />
H(ξ, ψ, η) = H(ξ, η), Ψ(ξ, ψ, η) = tan −1<br />
<br />
Hη(ξ, η) tan ψ − 3Hξ(ξ, η)<br />
.<br />
k<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 11 / 23
Contact transformations in logarithmic coordinates<br />
Contact form<br />
We write mappings on H 1 as<br />
τ = −e ξ (sin ψ dξ + 3 cos ψ dη).<br />
(ξ, ψ, η) ↦→ (Ξ(ξ, ψ, η), Ψ(ξ, ψ, η), H(ξ, ψ, η))<br />
To define a stretch map fk, 0 < k < 1, set<br />
Ξ(ξ, ψ, η) = kξ.<br />
Then the contact condition yields<br />
H(ξ, ψ, η) = H(ξ, η), Ψ(ξ, ψ, η) = tan −1<br />
<br />
Hη(ξ, η) tan ψ − 3Hξ(ξ, η)<br />
.<br />
k<br />
The stretch map is obtained by setting<br />
H(ξ, η) = η.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 11 / 23
Heisenberg stretch map<br />
Definition<br />
Let 0 < k < 1. The Heisenberg stretch map is given by<br />
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ<br />
( k ), η).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 12 / 23
Heisenberg stretch map<br />
Definition<br />
Let 0 < k < 1. The Heisenberg stretch map is given by<br />
Properties:<br />
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ<br />
( k ), η).<br />
fk is quasiconformal with fk(p)H = p k H ,<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 12 / 23
Heisenberg stretch map<br />
Definition<br />
Let 0 < k < 1. The Heisenberg stretch map is given by<br />
Properties:<br />
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ<br />
( k ), η).<br />
fk is quasiconformal with fk(p)H = p k H ,<br />
fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0),<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 12 / 23
Heisenberg stretch map<br />
Definition<br />
Let 0 < k < 1. The Heisenberg stretch map is given by<br />
Properties:<br />
fk(ξ, ψ, η) = (kξ, tan −1 tan ψ<br />
( k ), η).<br />
fk is quasiconformal with fk(p)H = p k H ,<br />
fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0),<br />
<br />
z<br />
f−1(z, t) = |z| 2 −t , −it |z| 4 +t2 <br />
is 1-QC inversion in the Korányi unit<br />
sphere with inversion relations<br />
dH(f−1(p), f−1(q)) = dH(p, q)<br />
pHqH<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
and f−1(p)H = 1<br />
.<br />
pH<br />
HCAA 2012 12 / 23
Analytic characterization of QC maps on H 1<br />
Theorem (Korányi, Reimann, Pansu, Vodop’yanov)<br />
A homeomorphism f : Ω → Ω ′ between domains in H 1 is quasiconformal if<br />
and only if<br />
f ∈ HW 1,4<br />
loc (Ω),<br />
f is a weakly contact map,<br />
Here,<br />
∃ 1 ≤ K < ∞ such that DHf (p) 4 ≤ KJ(p, f ) a.e.<br />
DHf 4 = (|ZfI | + |¯ZfI |) 4<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
and J(·, f ) = (|ZfI | 2 − |¯ZfI | 2 ) 2 .<br />
HCAA 2012 13 / 23
Analytic characterization of QC maps on H 1<br />
Theorem (Korányi, Reimann, Pansu, Vodop’yanov)<br />
A homeomorphism f : Ω → Ω ′ between domains in H 1 is quasiconformal if<br />
and only if<br />
f ∈ HW 1,4<br />
loc (Ω),<br />
f is a weakly contact map,<br />
Here,<br />
∃ 1 ≤ K < ∞ such that DHf (p) 4 ≤ KJ(p, f ) a.e.<br />
DHf 4 = (|ZfI | + |¯ZfI |) 4<br />
and J(·, f ) = (|ZfI | 2 − |¯ZfI | 2 ) 2 .<br />
Definition (Distortion)<br />
<br />
<br />
|ZfI<br />
K(·, f ) = <br />
| + |ZfI | <br />
<br />
<br />
|ZfI | − |ZfI |<br />
and Kf = K(·, f )∞.<br />
Note: H(·, f ) = K(·, f ) a.e.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 13 / 23
Proposition<br />
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion<br />
K((ξ, ψ, η), fk) = 1 + tan2 ψ<br />
k 2 + tan 2 ψ<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
1<br />
and Kfk = .<br />
k2 HCAA 2012 14 / 23
Proposition<br />
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion<br />
K((ξ, ψ, η), fk) = 1 + tan2 ψ<br />
k 2 + tan 2 ψ<br />
Let F be the class of QC maps<br />
1<br />
and Kfk = .<br />
k2 g : A(a, b) := {p ∈ H 1 : a < pH < b} → A(a k , b k )<br />
which map the inner and outer boundary components of A(a, b)<br />
homeomorphically to the respective boundary components of A(a k , b k ).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 14 / 23
Proposition<br />
The stretch map fk, 0 < k < 1 is a QC map of H 1 with distortion<br />
K((ξ, ψ, η), fk) = 1 + tan2 ψ<br />
k 2 + tan 2 ψ<br />
Let F be the class of QC maps<br />
1<br />
and Kfk = .<br />
k2 g : A(a, b) := {p ∈ H 1 : a < pH < b} → A(a k , b k )<br />
which map the inner and outer boundary components of A(a, b)<br />
homeomorphically to the respective boundary components of A(a k , b k ).<br />
Question<br />
Is fk extremal (i.e., a minimizer for maximal or mean distortion) in F?<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 14 / 23
Extremality of the stretch I (mean distortion)<br />
Theorem<br />
For all g ∈ F we have that<br />
<br />
K(p, fk)<br />
A(a,b)<br />
2 ρ0(p) 4 <br />
dµ(p) ≤<br />
with ρ0 = (log b<br />
Here <br />
Ω hρ4 0<br />
a )−1 ∇0 log( · H)0.<br />
dµ =<br />
Proof by modulus method.<br />
K(p, g)<br />
A(a,b)<br />
2 ρ0(p) 4 dµ(p)<br />
<br />
Ω hρ4 0 dµ<br />
<br />
Ω ρ4 0 dµ and µ denotes the Haar measure on H1 .<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 15 / 23
Modulus of curve families – a ‘quasi-invariant’ for QC maps<br />
Definition<br />
The 4-modulus of a curve family Γ on H1 is given by<br />
<br />
M4(Γ) = inf ρ<br />
ρ∈adm(Γ)<br />
4 dµ,<br />
adm(Γ) = {ρ : H 1 → [0, ∞] Borel with <br />
For γ = (z, t) horizontal: <br />
γ ρ dℓ = b<br />
a ρ(γ(s))|˙z(s)| ds.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
γ<br />
H 1<br />
ρ dℓ ≥ 1 for γ ∈ Γ loc. rectif.}.<br />
HCAA 2012 16 / 23
Modulus of curve families – a ‘quasi-invariant’ for QC maps<br />
Definition<br />
The 4-modulus of a curve family Γ on H1 is given by<br />
<br />
M4(Γ) = inf ρ<br />
ρ∈adm(Γ)<br />
4 dµ,<br />
adm(Γ) = {ρ : H 1 → [0, ∞] Borel with <br />
For γ = (z, t) horizontal: <br />
γ ρ dℓ = b<br />
a ρ(γ(s))|˙z(s)| ds.<br />
Cf. work by Markina and Vodop’yanov, and by Tang.<br />
Theorem<br />
γ<br />
H 1<br />
ρ dℓ ≥ 1 for γ ∈ Γ loc. rectif.}.<br />
Let f : Ω → Ω ′ be a QC map between domains in H1 and let Γ be a family<br />
of curves in Ω. Then<br />
<br />
M4(f (Γ)) ≤ K((z, t), f ) 2 ρ 4 (z, t) dµ(z, t) for all ρ ∈ adm(Γ).<br />
Ω<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 16 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
HCAA 2012 17 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 17 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 17 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2 ρ 4 0 dµ.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 17 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ.<br />
Thus,<br />
<br />
K(·, fk) 2 ρ 4 <br />
0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 17 / 23
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),<br />
Γ0 and fk are related in the following way (these are general conditions<br />
which guarantee “=” in the modulus-distortion estimate.):
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),<br />
Γ0 and fk are related in the following way (these are general conditions<br />
which guarantee “=” in the modulus-distortion estimate.):<br />
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:<br />
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),<br />
Γ0 and fk are related in the following way (these are general conditions<br />
which guarantee “=” in the modulus-distortion estimate.):<br />
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:<br />
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|<br />
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and<br />
thus <br />
f ◦γ ρ′ dℓ ≥ <br />
γ ρ dℓ with ρ′ ρ<br />
=<br />
||ZfI |−|¯ZfI || ◦ f −1 .<br />
<br />
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4 <br />
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),<br />
Γ0 and fk are related in the following way (these are general conditions<br />
which guarantee “=” in the modulus-distortion estimate.):<br />
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:<br />
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|<br />
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and<br />
thus <br />
f ◦γ ρ′ dℓ ≥ <br />
γ ρ dℓ with ρ′ ρ<br />
=<br />
||ZfI |−|¯ZfI || ◦ f −1 .<br />
<br />
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4 <br />
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ<br />
◮ K(·, fk) is constant along each γψ,η ∈ Γ0;
Choice of the curve family Γ0<br />
Motivation:<br />
Curves in Γ0 play the role of radial segments (Korányi, Reimann),<br />
Γ0 and fk are related in the following way (these are general conditions<br />
which guarantee “=” in the modulus-distortion estimate.):<br />
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:<br />
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|<br />
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and<br />
thus <br />
f ◦γ ρ′ dℓ ≥ <br />
γ ρ dℓ with ρ′ ρ<br />
=<br />
||ZfI |−|¯ZfI || ◦ f −1 .<br />
<br />
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4 <br />
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ<br />
◮ K(·, fk) is constant along each γψ,η ∈ Γ0;<br />
◮ integration formula:<br />
<br />
2 ln b<br />
h dµ =<br />
h(γψ,η(s))|˙zψ,η(s)| 4 <br />
ds 12 cos 2 (ψ)dψdη.<br />
A(a,b)<br />
Q<br />
2 ln a
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
HCAA 2012 19 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 19 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 19 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2 ρ 4 0 dµ.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 19 / 23
Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ.<br />
Thus,<br />
<br />
K(·, fk) 2 ρ 4 <br />
0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 19 / 23
Extremality of the stretch II (maximal distortion)<br />
M4(Γ) = π 2 ln <br />
b −3<br />
a<br />
(cf. Korányi, Reimann) and thus<br />
and M4(fk(Γ)) = k −3 π 2 ln <br />
b −3<br />
a ,<br />
M4(fk(Γ)) K 2<br />
fk M4(Γ).<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 20 / 23
Extremality of the stretch II (maximal distortion)<br />
M4(Γ) = π 2 ln <br />
b −3<br />
a<br />
(cf. Korányi, Reimann) and thus<br />
and M4(fk(Γ)) = k −3 π 2 ln <br />
b −3<br />
a ,<br />
M4(fk(Γ)) K 2<br />
fk M4(Γ).<br />
Question<br />
Is the stretch map fk extremal for the maximal distortion within F?<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 20 / 23
Extremality of the stretch II (maximal distortion)<br />
M4(Γ) = π 2 ln <br />
b −3<br />
a<br />
(cf. Korányi, Reimann) and thus<br />
and M4(fk(Γ)) = k −3 π 2 ln <br />
b −3<br />
a ,<br />
M4(fk(Γ)) K 2<br />
fk M4(Γ).<br />
Question<br />
Is the stretch map fk extremal for the maximal distortion within F?<br />
Proposition<br />
The stretch map fk is extremal for the maximal distortion in the class F0<br />
of C 1 orientation- and sphere-preserving maps g ∈ F which map the t-axis<br />
to the t-axis, i.e., Kfk ≤ Kg for all g ∈ F0.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 20 / 23
Extremality of the stretch II (maximal distortion)<br />
M4(Γ) = π 2 ln <br />
b −3<br />
a<br />
(cf. Korányi, Reimann) and thus<br />
and M4(fk(Γ)) = k −3 π 2 ln <br />
b −3<br />
a ,<br />
M4(fk(Γ)) K 2<br />
fk M4(Γ).<br />
Question<br />
Is the stretch map fk extremal for the maximal distortion within F?<br />
Proposition<br />
The stretch map fk is extremal for the maximal distortion in the class F0<br />
of C 1 orientation- and sphere-preserving maps g ∈ F which map the t-axis<br />
to the t-axis, i.e., Kfk ≤ Kg for all g ∈ F0.<br />
Proof idea: write contact transformation g ∈ F0 in logarithmic<br />
coordinates. Sphere-preserving: Ξ = Ξ(ξ), apply MVT: ∃ξ0 with<br />
Ξξ(ξ0) = k. Formula for distortion. Find p0 with K(p0, g) ≥ Kfk .<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 20 / 23
Thank you for your attention!<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 21 / 23
References<br />
A. Korányi and H. M. Reimann<br />
Horizontal normal vectors and conformal capacity of spherical rings in<br />
the Heisenberg group<br />
Bull. Sci. Math. (2), 111(1):3–21, 1987<br />
A. Korányi and H. M. Reimann<br />
Foundations for the theory of quasiconformal mappings on the<br />
Heisenberg group<br />
Adv. Math, 111(1):1–87, 1995<br />
I. D. Platis<br />
The geometry of complex hyperbolic packs<br />
Math. Proc. Cambridge Philos. Soc., 147(1):205–234, 2009<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 22 / 23
Return<br />
Figure: Curves in the Legendrian foliation of a Korányi sphere