A classification theorem for Helfrich surfaces

In this paper we study the functional $\mathcal W{} _{\lambda _1,\lambda _2}$ , which is the sum of the Willmore energy, $\lambda _1$ -weighted surface area, and $\lambda _2$ -weighted volume, for surfaces immersed in $\mathbb R ^3$ . 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 $\lambda _1\ge 0$ and small $L^2$ norm of tracefree curvature, with no assumption on the growth of the curvature in $L^2$ at infinity. This not only improves the gap lemma due to Kuwert and Schätzle for Willmore surfaces immersed in $\mathbb R ^3$ 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.