The fence conjecture for polygonal affine spheres David Dumas and Michael Wolf

In a recent paper (also on the arxiv), Michael Wolf and I study complete hyperbolic affine spheres in R3 that are asymptotic to convex polygonal cones (i.e. the cones over convex polygons in RP2). The part of the surface xyz=1 in the positive octant is an example of such a surface, in this case asymptotic to the cone over a triangle.

A key object of study in our work is a cubic form on the tangent bundle of the affine sphere, known as the Pick form. One way to think about the Pick form is that it measures the difference between notions of geodesic for two natural connections on the surface—the restriction of the flat connection of R3, and the Levi-Civita connection of a metric on the surface constructed from its Hessian.

For an affine sphere over a convex n-gon, we show that the Pick form vanishes at exactly (n-3) points—the Pick zeros. This set of Pick zeros is a projective invariant of the polygon.

We wondered whether anything about the arrangement of the Pick zeros could be understood directly in terms of projective geometry of the polygon, without referring to constructions in affine differential geometry.

In a series of computer experiments (see software below) we noticed that the Pick zeros seem to "avoid" certain parts of the polygon. To make this precise, consider the triangles formed by triples of consecutive vertices of a convex n-gon. Cutting all of these away leaves a smaller n-gon, which we call the fence of P. In some cases the fence degenerates to a point, as in a regular polygon, or to a line segment. Our experiments suggest the following:

Conjecture: For any convex n-gon in RP2, the Pick zeros lie inside the fence.
The image above shows a few examples: The five marked points (in red) in each octagon are the Pick zeros. The diagonals of each octagon which we cut along to get the fence are shown in blue, so the fence itself is the smaller octagon formed from the blue edges.

Animations

The animations below show the Pick zeros (red) and fence (blue) for various families of polygons. Each animation corresponds to a ceratin family of complex polynomials: As described in our paper, the Pick form of the affine sphere over a convex n-gon can be identified with a monic polynomial of degree (n-3), and conversely, the polynomial determines the convex polygon up to projective transformations. In some of the animations, the corresponding polynomial is also "shown" through the locations of its roots in the complex plane.

The animations are hosted on YouTube for convenient online viewing. Higher-quality MP4 video files are also available for download.

Octagon Dance

This family of octagons corresponds to polynomials with roots moving along five circles in the complex plane. Each circle contains the origin and one of the fifth roots of unity. The roots move around the circles at different rates, resuling in a family that visits various parts of the moduli space. The animation starts and ends with the regular octagon, corresponding to the polynomial z5.

Octagon Dance (dual view)

This variation on the previous animation shows side-by-side views of the zeros of the complex polynomial and the corresponding polygon with its Pick zeros.

Heptagon Cycle

This family of heptagons corresponds to polynomials of the form (z4 - c) with the complex constant c starting at the origin, moving through positive real values, and then rotating along the unit circle. In the corresponding heptagons, the Pick zeros appear to approach and then march along the fence, turning corners as necessary.

Heptagon Cycle (dual view)

This variation on the previous animation shows side-by-side views of the zeros of the complex polynomial and the corresponding polygon with its Pick zeros.

Software

The main software component in these experiments is a blaschke, an open-source Python package for computing the Blaschke metric, affine frame field, and real projective polygon corresponding to a complex polynomial. Images of the polygons, Pick zeros, and fences were created using the pycairo drawing library and matplotlib. The video files were encoded with ffmpeg using the libx264 implementation of h264.

Older Experiments

These animated GIFs were created in an earlier stage of our experimental work. They show the polygons (red), fence (green), and Pick zeros (blue) for several families of polynomial cubic differentials. Thus, for these images, the fence conjecture says that the blue points never cross the green lines.

• fence-pent1.gif   -   The family z2 - c for c in [-0.8,0.8]

• fence-pent2.gif   -   The family eit(z2 - 0.5) for t in [0,2 π]

• fence-pent3.gif   -   The family eit(z2 - 0.7) for t in [0,2 π]

• fence-hex1.gif   -   A family of monic cubic polynomials with root sum zero; two of the roots move in circles at different speeds.
These older animations were created using a different method for computing the polygon and its Pick zeros, and this method results in significant numerical error. Sometimes this error can be seen in the "jittery" motion of the Pick zeros and the vertices.

Revision history

• 2014-03-20   —   Initial post with animated GIFs
• 2015-01-17   —   Major update with new computation method, videos, and code release

Acknowledgement

This material is based upon work supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.