AHLFORS-REGULAR CURVES
Zinsmeister Michel,
MAPMO, Université d’Orléans
Ahlfors Centennial
Celebration,Helsinki,
August 2007
1.INTRODUCTION
L
5r
r
The Cauchy operator on L
Is defined as
Calderon’s question:
when is this operator
bounded on L2(ds)?
Ahlfors-regularity :
Theorem (G.David): The Cauchy operator is bounded on
L2 for all Ahlfors-regular curves.
Oberwolfach, 1987
Equivalent definitions:
An Ahlfors-regular curve need not be a Jordan arc: if we
ask the curve to be moreover a quasicircle we get an
interesting class of curves.
z2
z1
A curve passing through infinity is
said to be Lavrentiev or chord-arc if
there exists a constant C>0 such that
for any two points of the curve the
length of the arc joining the two
points is bounded above by C times
the length of the chord.
Ahlfors-regularity+quasicircle=Chord-arc
Theorem (Z): If U is a simply connected domain whose
boundary is Ahlfors-regular and f is the Riemann map
from the upper half-plane onto U then b= Log f’ is in
BMOA. Moreover if AR denotes this set of b’s, the interior
of AR in BMOA is precisely the set of b’s coming from
Lavrentiev curves.
Theorem (Pommerenke): If b is in BMOA with a small
norm then b=Log f’ for some Riemann map onto a
Lavrentiev curve
These two theorems suggest the possibility of a specific
Teichmüller theory.
2. BMO-TEICHMÜLLER THEORY
2.1 SOME FACTS FROM CLASSICAL TEICHMÜLLER
THEORY
Let S be a hyperbolic Riemann surface and f,g two
quasiconformal homeomorphisms from S to T,U
respectively:
f
T
g
U
S
We say that f,g are equivalent if gof-1 is homotopic
modulo the boundary to a conformal mapping.
The Teichmüller space T(S) is the set of equivalence
classes of this relation.
The maps f,g can be lifted to qc homeomorphisms F,G of
the upper half plane H, the universal cover of T,U.
H
F
H
T
S
f
f and g are
equivalent iff F-1oG
restricted to R is
Möbius.
Notice that E(h)(z)=h(z) if h is Möbius.
H
L
Welding:
2.2. BMO-TEICHMÜLLER THEORY
In order to develop this theory we need some definitions:
We wish to construct a Teichmüller theory
corresponding to absolutely continuous weldings.
Using a theorem of Fefferman-Kenig-Pipher we recognize
the natural candidate as follows:
The problem of finding conditions ensuring absolute continuity has a long
history starting with Carleson and culminating with a theorem by Fefferman,
Kenig and Pipher.
As in the classical theory we wish to identify with a space
of quasisymmetries and a space of quadratic
differentials.
The fact that the map is into follows from F-K-P theorem
To prove that it is onto we first consider the « universal »
case , i.e. the case S=D.
We wish now to have a nice Bers embedding for the
restricted Teichmüller spaces:
A geometric charcterisation of domains such that Log f’ is
in BMOA has been given by Bishop and Jones.
The boundary of such domains may have Haudorff
dimension >1 so this class is much larger than AR.
Question: Is the subset L corresponding to
Lavrentiev curves connected?
3. RECTIFIABILITY AND GROWTH
PROCESSES
3.1 Hastings-Levitov process
These curves are obtained by iteration of simple
conformal maps
Fix d>0 and consider fd the conformal map sending the
complement of the unit disc to the complement of the
unit disc minus the segment [1,1+d] with positive
derivative at infinity.
fd
This mapping is completely explicit and in particular
fn-1
The diameter of the nth cluster increases exponentially
We normalize the mapping fn by dividing by the z-term
and then substracting the constant one.
Let S0 denote the set of univalent fiunctions on the
outside of the unit disk of the form z+a/z+..
The random process we have constructed induces a
probability measure Pn on S0.
Theorem (Rohde, Z): the proces has a scaling limit in the
sense that the sequence Pn has a weak limit P as n goes
to infinity.
Theorem (Rohde,Z): If d is large enough, P-as the length
of the limit cluster is finite.
3.2 Löwner processes
We consider the Löwner differential equation:
Marshall and Rohde have shown that if the driving
function is Hölder-1/2 continuous with a small norm
then gs maps univalently the unit disc onto the disc
minus a quasi-arc.
Problem: find extra condition on the driving function
so that the quasi-arc is rectifiable.
Theorem (Tran Vo Huy, Nguyen Lam Hung, Z.): It is
the case if the driving function is in the Sobolev
space W1,3 with a small norm.
Idea of proof:
Problem: the derivative at 0 of these maps is 0