A classification theorem for Helfrich surfaces

TL;DR

This paper classifies all smooth immersed critical points of a Helfrich-type energy functional in 3, revealing conditions under which such surfaces exist or do not exist, especially with small tracefree curvature.

Contribution

It provides a complete classification of critical points of the Helfrich functional with nonnegative surface tension and small tracefree curvature, including non-existence results.

Findings

Classified all smooth immersed critical points for the Helfrich functional.

Proved non-existence of critical points with positive surface area and volume weights.

Established conditions for the existence and non-existence of Helfrich surfaces.

Abstract

In this paper we study the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-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 $\lambda_1\ge0$ 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.