AHLFORS-REGULAR CURVES

Ahlfors Centennial Celebration,Helsinki,August 2007

Zinsmeister Michel,

MAPMO, Université d’Orléans

1.INTRODUCTION

L

Calderon’s question: when is this operator bounded on L2(ds)?

The Cauchy operator on LIs defined asr

5r

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.

z1

z2

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:

S

T

U

f

g

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.

S T

H H

f

Ff 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