A classification theorem for Helfrich surfaces

In this paper we study the functional , which is the sum of the Willmore energy, -weighted surface area, and -weighted volume, for surfaces immersed in . 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 and small norm of tracefree curvature, with no assumption on the growth of the curvature in at infinity. This not only improves the gap lemma due to Kuwert and Schatzle for Willmore surfaces immersed in but also implies the non-existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted.