You are currently browsing the tag archive for the ‘quasiconformal mapping’ tag.
Kari Astala, Steffen Rohde, Eero Saksman and I have (finally!) uploaded to the arXiv our preprint “Homogenization of iterated singular integrals with applications to random quasiconformal maps“. This project started (and was largely completed) over a decade ago, but for various reasons it was not finalised until very recently. The motivation for this project was to study the behaviour of “random” quasiconformal maps. Recall that a (smooth) quasiconformal map is a homeomorphism that obeys the Beltrami equation
The random expression (2) is somewhat reminiscent of a moment of a random matrix, and one can start computing it analogously. For instance, if one has a decomposition such as (1), then (2) expands out as a sum
If all the were distinct then one could use independence to factor the expectation to get
In principle all of these limits are computable, but the combinatorics is remarkably complicated, and while there is certainly some algebraic structure to the calculations, it does not seem to be easily describable in terms of an existing framework (e.g., that of free probability).
Previous set of notes: Notes 1. Next set of notes: Notes 3.
We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map from an open subset
of the complex plane to another open set
is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of
to be nowhere vanishing. We then say that the two open sets
are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation
we see that they almost preserve small circles, indeed for
small the circle
will approximately map to
.
In previous quarters, we proved a fundamental theorem about this concept, the Riemann mapping theorem:
Theorem 1 (Riemann mapping theorem) Let
be a simply connected open subset of
that is not all of
. Then
is conformally equivalent to the unit disk
.
This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps from
to
that map some fixed point
to
, pick one that maximises the magnitude
of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If
avoids some point in
, one can compose
with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase
without destroying injectivity; see the previous lecture notes for details. The conformal map
is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points
in
, and requiring
to be zero and
to be positive real.
It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).
To make the above discussion more precise we need some notation.
Definition 2 (Circle packing) A (finite) circle packing is a finite collection
of circles
in the complex numbers indexed by some finite set
, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices
are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)
It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.
The first basic theorem in the subject is the following converse statement:
Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.
Among other things, the circle packing theorem thus implies as a corollary Fáry’s theorem that every planar graph can be drawn using straight lines.
Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:
Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.
Exercise 5 Let
be a connected planar graph with
vertices. Show that the following are equivalent:
- (i)
is a maximal planar graph.
- (ii)
has
edges.
- (iii) Every drawing
of
divides the plane into faces that have three edges each, and each edge is adjacent to two distinct faces. (This includes one unbounded face.)
- (iv) At least one drawing
of
divides the plane into faces that have three edges each, and each edge is adjacent to two distinct faces.
(Hint: you may use without proof Euler’s formula
for planar graphs, where
is the number of faces including the unbounded face.)
Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:
Conjecture 6 (Informal Thurston conjecture) Let
be a simply connected domain, with two distinct points
. Let
be the conformal map from
to
that maps
to the origin and
to a positive real. For any small
, let
be the portion of the regular hexagonal circle packing by circles of radius
that are contained in
, and let
be an circle packing of
with the same nerve (up to isomorphism) as
, with all “boundary circles” tangent to
, giving rise to an “approximate map”
defined on the subset
of
consisting of the circles of
, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that
is zero and
is a positive real. Then
converges to
as
.
A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation , while (sufficiently smooth) quasiconformal maps only obey an inequality
. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map
alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).
Recent Comments