A study of surface theory in conformal 3-space, with an application to the extremals of the Willmore functional, which can be thought of as the conformal area of a surface in this geometry.
Among the results are a proof that every compact extremal of genus 0 is conformally a minimal surface. This relies on a vanishing theorem plus a careful analysis of the singularities of the geometry near the `umbilic' points. Also the critical values of the Willmore functional on 2-spheres are shown to be discrete and the moduli space of the extrema having the first non-minimal critical value is computed.
Since this paper, much has been done. For an update, see Surfaces in Conformal Geometry.
Lucas Hsu kindly compiled a list of errata and has allowed me to include an amstex version of it here.
Reprints are available.