A classification theorem for Helfrich surfaces

Abstract

In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the 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. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.

21 pages

References in corpus (1)

• Surface diffusion flow near spheres

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