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.

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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One Response 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

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