Course progress

This is a not very detailed record on what happened in each course meeting. There should be course recordings in 北大教学网.

Sep 08

We discussed the definition of Teichmuller space, and the differentiable structure of $\mathrm{Teich}(T^2)$.

Reference: [FM], pages 263-269.

Sep 10

We discussed the algebraic topology of $\mathrm{Teich}(S_g)$, by identifying it with $\mathrm{DF}(\pi_1(S_g),PSL(2,\mathbb R))/PGL(2,\mathbb R)$.

Then we show that $\mathrm{Teich}(S_0^3)\cong\mathbb R^3_+$, by considering the lengths of the three boundaries.

Any closed surface can be decomposed into pairs of pants. We defined twisting parameters using seams.

Reference: [FM], pages 269-282.

Sep 15

We discussed Fenchel-Nielson coordinates for Teichmuller space, and in particular, showed that $\mathrm{Teich}(S_g)\cong \mathbb R^{3g-3}_+\times \mathbb R^{3g-3}$.

Teichmuller space can be regarded as the space of marked hyperbolic surfaces, or as the space of marked complex structures. We introduced quasi-conformal maps, and stated Teichmuller’s result on the Teichmuller metric using quasi-conformal maps.

Reference: [FM], pages 282-287, 294-300.

Sep 22

We discussed measured foliations and quadratic differentials $\mathrm{QD}(X)$ on a Riemann surface, possibly bordered and punctured. These two concepts are closed related and essentially equivalent. We defined Teichmuller mappings based on the definition of quadratic differentials. We also computed $\mathrm{dim}_{\mathbb C} \mathrm{QD}(X)=3g-3$.

Reference: [FM], pages 300-305, 309-312.

Sep 24

As an example, we studied the Swiss Cross as a construction of a genus $2$ Riemann surface together with a quadratic differential. We stated the Teichmuller existence and uniquess theorem, and explained the diffeomorphism map between $\mathrm{QD}_1(X)$ and $\mathrm{Teich}(S)$. The Teichmuller metric is a complete metric defined by the dilatation of the Teichmuller map relating two conformal structures.

Reference: [FM], pages 313-315, 320-323, 330-332, 337-341.

Oct 15

We introducted mapping class groups, and discussed several examples (disk, annulus, torus with zero or $1$ marked points, $S_{0,3}$ and $S_{0,4}$.) We introduced the concept of the moduli space of (smooth) conformal/hyperbolic structures, and discussed the example of $\mathcal M(T^2)$.

Reference: [FM], Chapter 2, pages 342-349.