  A domain in a Langmuir monolayer can be expected to have a shape that
reflects the textural anisotropy of the material it contains. This paper
explores the consequences of XY-like ordering. It is found that an extension of
the Wulff construction allows for the calculation of two-dimensional domain
shapes when each segment of the perimeter has an energy that depends both on
its orientation and its location. This generalized Wulff construction, and
newly-derived exact expressions for the order parameter texture in a circular
domain, lead to results for the shape of a large domain. The most striking
result is that under general conditions such domains will inevitably develop
cusps. We show that the onset of a cusp is mathematically related to a phase
transition. The present approach is equivalent to a Landau mean-field version
of the theory.
