A classification theorem for Helfrich surfaces

In this paper we study the functional W , which is the the sum of the Willmore energy, weighted surface area, and weighted volume, for surfaces immersed in R^3. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with nonnegative surface area weight and small L^2 norm of tracefree curvature. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.