Undergraduate Research at GANG
Summer 2002

Spherical Polygons

A spherical poylgon is a loop of geodesic segments on a 2-sphere, each of whose side lengths is between 0 and the semicircumference inclusively.

We prove that the side lengths of a spherical polygon satisfy the spherical polygon inequalities. And conversely, given any lengths which satisfies these, there exists a spherical polygon whose sides have these lengths.

Given n unitary matrices Mk in SU2, with M1... Mn = 1, the n-gon inequalities are necessary conditions for the simultaneous unitarizability of M1, ... , Mn.

Conversely, if M1, M2, M3 in SL2(C) with M1 M2 M3=1 are individually unitarizable and irreducible, and nk in [0, 1/2] are defined by cos 2 pi nk = 1/2 tr Mk, then the spherical 3-gon inequalities imply that M1, M2, M3 are simultaneously unitarizable. This converse is false for n >3.

This problem has application to the construction of the moduli of constant mean curvature genus zero surfaces with n Delaunay ends (n-noids), where the asymptotic necksizes of the ends must satisfy the n-gon inequalities.

See our paper below for proofs of the theorems and further details.


Spherical Polygons explicates our results on the spherical polygon inequalities and simultaneous unitarizability.