# Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will appreciate.

Qn. 1. For considering a minimal dilatation map between open annuli $A(1,r_1)$ and $A(1,r_2)$,why don't we consider a certain homotopy class like the general case of Teichmuller's theorem, is it because that all maps between two annuli are homotopic ? Between $f(x)$ and $g(x)$ we construct a homotopy by composition of two homotopies : first radially adjusting $g(x)$ such that $|f(x)|= |g(x)|$ and then rotating the image of $g(x)$ under the first homotopy along the circle of radius $|f(x)|$ to match with $f(x)$ by the second homotopy ?

Qn. 2. a) Suppose we want two construct a minimal dilatation map between two open ( without boundary ) pair of pants , i.e. 2-sphere minus 3 disjoint closed disks, do we have to look at the minimal dilatation map between certain pair of pants with geodesic boundary of specified boundary lengths which should correspond to the open PPs such that the extremal length of certain curve families ( probably the ones joining the boundaries ) remains invariant by this open surface-to-compact-surface-with-boundary transition ? [ I am just trying to imitate the arguments for the case of annuli ]

b) If the answer to a) is yes, then intuitively can one guess that the minimal dilatation map between the "corresponding" pair of pants with geodesic boundaries ( say the first has longer boundaries: so "fat" PP , the other has shorter boundaries , so it is "thin" PP ) is obtained by "stretching the fat PP along " the common orthogonal to geodesic boundaries ? Is there a more explicit way , like the annuli, to describe this minimal dilation map ?

Qn. 3. I accept the result without proof ( by using Riemann-Roch , mentioned F. Gardiner's book Teichmuller Theory and Quadratic Differentials, P.26, Ch. 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n$ . Now for open annulus $A$, $g=0, m=2, n=0$, we get $dim_RQD(X)=0$ ! I am a bit puzzled why it is zero ! What should be the genus of an open annulus in any case, shouldn't it be zero ? And for q.diffs on annulus $A$, should we look at $q=\phi(z)dz^2$ when $\phi$ is a function on the annulus embedded in complex plane or should we lift it to upper half plane and consider the $\phi(z)$ with $\phi(z) = \phi(\gamma(z))\frac{\bar{\gamma'(z)}} {\gamma'(z)}$ for all $\gamma \in Deck(H/A)$ ? I guess the second approach makes more sense ?

Qn. 4. $FOR REFERENCES :$ I recently finished studying Lipman Ber's paper " Q.C. maps and Teichmuller's theorem " , which proves the uniqueness and existence for closed Riemann surfaces. Is there any good references/books/research paper about Teichmuller's theorem for punctured ( with cusps ) and open Riemann surfaces ?

I am sorry for this very long question if that bothers you.

• It may be better to split this up into several questions rather than grouping all 4 (with subparts) together in one. It is ultimately tidier I think. – Steven Gubkin Feb 24 '11 at 6:36
• Good suggestion ! – Analysis Now Feb 24 '11 at 13:25

## 1 Answer

I don't have time to answer your substantive questions, but I can recommend two very good sources for your 4th question :

1. Quasiconformal maps & Teichmüller theory by Fletcher-Markovich
2. Teichmüller Theory by John Hubbard