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AHLFORS-REGULAR CURVES
Zinsmeister Michel, MAPMO, Université dOrléans
Ahlfors Centennial Celebration,Helsinki,August
2007
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1.INTRODUCTION
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L
5r
The Cauchy operator on L Is defined as
r
Calderons question when is this operator
bounded on L2(ds)?
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Ahlfors-regularity
Theorem (G.David) The Cauchy operator is bounded
on L2 for all Ahlfors-regular curves.
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Oberwolfach, 1987
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Equivalent definitions
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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.
A curve passing through infinity is said to be
Lavrentiev or chord-arc if there exists a
constant Cgt0 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.
z2
z1
Ahlfors-regularityquasicircleChord-arc
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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 bs, the interior of AR in BMOA is
precisely the set of bs coming from Lavrentiev
curves.
Theorem (Pommerenke) If b is in BMOA with a
small norm then bLog f for some Riemann map
onto a Lavrentiev curve
These two theorems suggest the possibility of a
specific Teichmüller theory.
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2. BMO-TEICHMÜLLER THEORY
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2.1 SOME FACTS FROM CLASSICAL TEICHMÜLLER THEORY
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Let S be a hyperbolic Riemann surface and f,g two
quasiconformal homeomorphisms from S to T,U
respectively
T
f
S
U
g
We say that f,g are equivalent if gof-1 is
homotopic modulo the boundary to a conformal
mapping.
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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.
F
H
H
f and g are equivalent iff F-1oG restricted to R
is Möbius.
T
S
f
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Notice that E(h)(z)h(z) if h is Möbius.
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H
L
Welding
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2.2. BMO-TEICHMÜLLER THEORY
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In order to develop this theory we need some
definitions
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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
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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.
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As in the classical theory we wish to identify
with a space of quasisymmetries and a space of
quadratic differentials.
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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 SD.
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We wish now to have a nice Bers embedding for the
restricted Teichmüller spaces
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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 gt1 so this class is much larger than
AR.
Question Is the subset L corresponding to
Lavrentiev curves connected?
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3. RECTIFIABILITY AND GROWTH PROCESSES
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3.1 Hastings-Levitov process
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These curves are obtained by iteration of simple
conformal maps
Fix dgt0 and consider fd the conformal map sending
the complement of the unit disc to the complement
of the unit disc minus the segment 1,1d with
positive derivative at infinity.
fd
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This mapping is completely explicit and in
particular
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fn-1
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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 za/z..
The random process we have constructed induces a
probability measure Pn on S0.
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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.
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3.2 Löwner processes
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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.
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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.
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Idea of proof
Problem the derivative at 0 of these maps is 0
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