A classification theorem for Helfrich surfaces
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Mathematische Annalen
A classification theorem for Helfrich surfaces James McCoy · Glen Wheeler
Received: 19 January 2012 / Revised: 4 April 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper we study the functional W λ1 ,λ2 , which is the sum of the Willmore energy, λ1 -weighted surface area, and λ2 -weighted volume, for surfaces immersed in R3 . 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 λ1 ≥ 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 R3 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. Mathematics Subject Classification (2000)
35J30 · 58J05 · 35J62
1 Introduction Consider a surface Σ immersed in R3 via a smooth immersion f : Σ → R3 . There are several functionals of f relevant to our study: Financial support for G. Wheeler from the Alexander-von-Humboldt Stiftung is gratefully acknowledged. J. McCoy Institute for Mathematics and Its Applications, University of Wollongong, Northfields Ave, Wollongong, NSW 2522, Australia e-mail: [email protected] Present Address: G. Wheeler (B) Institute for Mathematics and Its Applications, University of Wollongong, Northfields Ave, Wollongong, NSW 2522, Australia e-mail: [email protected] G. Wheeler Institut für Analysis und Numerik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany
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J. McCoy, G. Wheeler
μ(Σ) =
dμ, the surface area Σ
1 f ν dμ, thesigned enclosed volume 3 Σ 1 |H |2 dμ, the Willmore energy W( f ) = 4 Σ 1 c0 (H − c0 )2 dμ + λ1 μ(Σ) + λ2 VolΣ, the Helfrich energy. Hλ1 ,λ2 ( f ) = 4 VolΣ = −
Σ
In the above we have used ·· to denote the inner product in R3 , ν to denote the inward-pointing unit normal, dμ to denote the area element induced by f on Σ, H to denote the mean curvature, and c0 , λ1 , λ2 to denote real numbers. When f : Σ → R3 is not an embedding, we still require the definition of the enclosed volume to make sense. For this reason we have used the expression above, which in the case of an embedding agrees with the measure of the interior via the divergence theorem, and is standard (see [10] for example). Our notation is further clarified in Sect. 2. The Helfrich functional Hcλ01 ,λ2 is of great interest in applications. Although it appears to have first been studied by Schadow [19], its modern form and subsequent popularity is due to Helfrich [12], just as the Willmore functional W := H00,0 was considered by Germain [3,9], Poisson [17], Thomsen [21], and others, long before the time of Willmore [23]. Helfrich famously proposed that the minimisers of the functional model the shape of an elastic lipid bilayer, such as a biomembrane. Since this time the model ha
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