chapter are [2, 23, 30, 71]. However, our treatment here differs in some ways,
most significantly in that it freely uses Brownian motion. We start with simple
connectedness and a proof of the Riemann mapping theorem. Although this is
really a first-year topic, it is so important to our discussion that it is included
here. The next section on univalent functions follows closely the treatment in
[30]; since the Riemann mapping theorem gives a correspondence between simply
connected domains and univalent functions on the disk it is natural to study the
latter. We then discuss two kinds of capacity, logarithmic capacity in the plane
which is classical and a "half-plane" capacity that is not as well known but similar
in spirit. Important uniform estimates about certain conformal transformations
are collected here; these are the basis for Loewner differential equations. Extremal
distance (extremal length) is an important conformally invariant quantity and is
discussed in Section 3.7. The next section discusses the Beurling estimate, which is a
corollary of a stronger result, the Beurling projection theorem. This is used to derive
a number of estimates about conformal maps of simply connected domains near the
boundary; what makes this work is the fact that the boundary of a simply connected
domain is connected. The final section discusses annuli, which are important when
considering radial or whole-plane processes.
Chapter 4 discusses the Loewner differential equation. We discuss three types,
chordal, radial, and whole-plane, although the last two are essentially the same.
It is the radial or whole-plane version that Loewner [66] developed in trying to
study the Bieberbach conjecture and has become a standard technique in conformal
mapping theory. The chordal version is less well known; Schramm [76] naturally
came upon this equation when trying to find a continuous model for loop-erased
walks and percolation. The final three sections deal with technical issues concerning
the equation. When does the solution of the Loewner equation come from a path?
What happens when solutions of the Loewner equation are mapped by a conformal
transformation? What does it mean for a sequence of solutions of the Loewner
equation to converge? The second of these questions is relevant for understanding
the relationship between the chordal and radial Loewner equations.
In Chapter 5 we return to Brownian motion. Some of the most important con-
formally invariant measures on paths are derived from complex Brownian motion.
After discussing a number of well-known measures (with perhaps a slightly different
view than usual), we discuss some important measures that have arisen recently:
excursion measure, Brownian boundary bubble measure, and the loop measure.
The Schramm-Loewner evolution (SLEK), which is the Loewner differential
equation driven by Brownian motion, is the topic of Chapter 6. With the Brow-
nian input, the Loewner equation becomes an equation of Bessel type, and much
of the analysis of SLE comes from studying such stochastic differential equations.
For example, the different "phases" of SLE (simple/non-simple/space-filling) are
deduced from properties of the Bessel equation. We discuss two important values
of the parameter n: K 6 which satisfies the locality property and K 8/3 which
satisfies the restriction property. One of the main reasons SLE has been so useful is
that crossing probabilities (Cardy's formula), crossing exponents, and other deriv-
ative exponents can be calculated exactly. These correspond to critical exponents
for lattice models. In the case K = 6, which corresponds to (among other things)
the limit of critical percolation, there is a particularly nice relationship between
SLE and Brownian motion that is most easily seen in an equilateral triangle.
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