  Determining the equilibrium configuration and shape of curved two-dimensional
films with (generalized) liquid crystalline (LC) order is a difficult infinite
dimensional problem of direct relevance to the study of generalized
polymersomes, soft matter and the fascinating problem of understanding the
origin and formation of shape (morphogenesis). The symmetry of the free energy
of the LC film being considered and the topology of the surface to be
determined often requires that the equilibrium configuration possesses singular
structures in the form of topological defects such as disclinations for nematic
films. The precise number and type of defect plays a fundamental role in
restricting the space of possible equilibrium shapes. Flexible closed vesicles
with spherical topology and nematic or smectic order, for example, inevitably
possess four elementary strength $+1/2$ disclination defects positioned at the
four vertices of a tetrahedral shell. Here we address the problem of
determining the equilibrium shape of flexible vesicles with generalized LC
order. The order parameter in these cases is an element of $S^1/Z_p$, for any
positive integer $p$. We will focus on the case $p =3$, known as triatic LCs.
We construct the appropriate order parameter for triatics and find the
associated free energy. We then describe the structure of the elementary
defects of strength $+1/3$ in flat space. Finally, we prove that sufficiently
floppy triatic vesicles with the topology of the 2-sphere equilibrate to
octahedral shells with strength $+1/3$ defects at each of the six vertices,
independently of scale.
