Quasiconformal Maps

Ok, time to get back to Riemann surfaces! We’ve been all about hyperbolic surfaces, and so first let’s compare the two of them: every oriented hyperbolic surface is a Riemann surface, and the conformal class of a complex structure contains a unique hyperbolic metric.

This fairly straightforward result (the first part is the fact that orientation preserving isometries of the hyperbolic plane are biholomorphic and so define a complex structure, and the second is from the uniformization theorem) is why the moduli of curves (complex structures on a surface) is just the moduli of marked hyperbolic surfaces (Teichmüller space) after getting rid of the marking (quotient by the mapping class group).

So now, we’ll start looking at the geometry of Teichmüller space.  One of the main tools there is the notion of a quasiconformal map.

Definition: Let $X$ and $X'$ be Riemann surfaces.  Let $K\geq 1$.  Then a map $F:X\to X'$ is $K$-quasiconformal if $F$ is a homeomorphism with continuous differential of maximal rank outside of a finite set $\Sigma$ of points, and we assume that $|\bar{\partial} F|\leq k|\partial F|$ outside of $\Sigma$, with $0\leq k<1$ and $K=\frac{1+k}{1-k}$.

What this means is that away from $\Sigma$, the differential takes a circle to an ellipse in the tangent spaces, and the ratio of the lengths of the axes is bounded by $K$.

So what can we do with this? Next time, we’ll really get into a big application of quasi-conformal maps and quadratic differentials, but now, we’ll just look at a couple of basic properties.

We can use quadratic differentials to build some special quasi-conformal maps.  Let $q$ be a quadratic differential.  It gives us a singular Euclidean metric, and we can scale it.  In the horizontal direction, we scale by $\exp(t/2)$ and in the vertical direction by $\exp(-t/2)$, which gives us a quasiconformal map $X\to X_t$ with constant $\exp(t)$, and we call these maps Teichmüller maps with initial differential $q$ and stretch factor $\exp(t/2)$, and they’re of constant area.

This lets us compute:

Theorem: Let $f:X\to X'$ be a quasiconformal map which is homotopic to a Teichmuller mapping with initial differential $q$ and stretch factor $L$.  Then $\int_X |f_x|dA\geq L||q||$ where $f_x$ is the derivative of $f$ in the horizontal direction and the norm is with respect to the singular metric from $q$.

This can be used to classify annuli.  An annulus is $A_a=[0,1]\times [0,a]/\sim$ where $(0,t)\sim (1,t)$ for all $t$.  Then for $b\geq a$, the smallest quasiconformal dilation $A_a\to A_b$ is $b/a$.

This, then, implies that $A_a\cong A_b$ if and only if $a=b$.

We’ll close with a theorem of Wolpert:

Let $X$ and $X'$ be hyperbolic surfaces and let $f:X\to X'$ be a $K$-quasiconformal map.  Then for any simple closed curve $c$, we have $\frac{\ell_c(X)}{K}\leq \ell_c(X')\leq K\ell_c(X)$, where $\ell_c$ is the length as we discussed before.  So that gives a nice interpretation of the $K$ in quasiconformal maps: it’s a bound on how much simple closed curves change their length. Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in Uncategorized. Bookmark the permalink.

3 Responses to Quasiconformal Maps

1. giulio says:

Dear Charles,

First of all, let me thank you for this amazing serie, it is very well written and pleasant to read.

I am a physicist and I got interested in the study of the moduli space of curves of genus zero because of its connections with quantum field theory.

In particular, I know that the moduli space has certain “factorization properties”, that is in order to compactify it you have to add codimension one strata which are written as products of other moduli spaces : $\partial M_{0,n} \superset M_{0,k} \times M_{0,n-k+2}$

Do you think it would be possible to find a meromorphic form $\omega_n$ of maximum rank such that its divisor is made of exactly these strata and its residue at $M_{0,k} \times M_{0,n-k+2}$ is written as $\omega_k \wedge \omega_{n-k+2}$?

I apologize if the question is ill-posed and lacks of rigour…I am just an humble physicist…

Thank you,
Giulio

2. Anonymous says:

,

3. prof drd horia orasanu says:

,is dignified to remarks that quasiconformal maps is fundamental in many codomains as observed prof drd horia orasanu and prof dr mircea orasanu and confirmed by prof dr Constantin Udriste as observed bottom where is uniform in space, but can vary in time. In fact, the time variation of can be eliminated by adding the appropriate function of time (but not of space) to the velocity potential, . Note that such a procedure does not modify the instantaneous velocity field derived from . Thus, the previous equation can be rewritten
(4.96)
where is constant in both space and time. Expression (4.96) is a generalization of Bernoulli’s theorem (see Section 4.3) that takes non-steady flow into account. However, this generalization is only valid for irrotational flow. For the special case of steady flow, we get
(4.97)
which demonstrates that for steady irrotational flow the constant in Bernoulli’s theorem is the same on all streamlines. (SeKelvin Circulation Theorem
According to the Kelvin circulation theorem, which is named after Lord Kelvin (1824-1907), the circulation around any co-moving loop in an inviscid fluid is independent of time. The proof is as follows. The circulation around a given loop is defined
(4.78)