# Bers slices and related galleries

This page provides a brief overview of my galleries of mathematical images and animations related to CP1 structures on surfaces. Further discussion of the underlying mathematics can be found in my research articles.

### Bers Slices of P(S)

The space P(S) of all marked complex projective structures on a surface S is a bundle of over the Teichmüller space T(S). The fiber over X can be identified with the vector space Q(X) of integrable holomorphic quadratic differentials on X.

When S is a punctured torus, P(S) has complex dimension two and the fibers over T(S) are complex lines. The images in the galleries below show parts of these fibers, where colors are used to distinguish CP1 structures with discrete holonomy representations from those with non-discrete holonomy. Animations are created by gradually changing the parameters of a 2-dimensional slice (such as the underlying complex structure and the viewing window within the fiber) to produce a sequence of frames.

### Ungrafting and Pleating rays

Using numerical integration of the Schwarzian differential equation, it is possible to make explicit computations relating to pleating rays in certain symmetric cases. (Specifically, the pleating ray in the Bers slice of a punctured torus is a straight line whenever the associated geodesic is invariant under an orientation-reversing isometry of the base hyperbolic structure.)

The image below was created with a Mathematica program.

• Hex torus - Length vs. ungrafting (bending) angle along the pleating ray in the hex torus bers slice for one of the three shortest curves.

### Polar Coordinates for the Bers Slice

McMullen has shown that the (normalized) length and slope of the bending lamination of the convex core boundary provide polar coordinates for the standard component in the Bers slice. The following picture displays the radial coordinate along a family of rays. The rays are colored according to the height of the associated rational slope.