Skip to main content

A classification theorem for Helfrich surfaces

Journal Article


Download full-text (Open Access)

Abstract


  • In this paper we study the functional W , which is the the sum of the Willmore energy, weighted surface area, and 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 nonnegative surface area weight 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.

Publication Date


  • 2013

Citation


  • McCoy, J. & Wheeler, G. E. (2013). A classification theorem for Helfrich surfaces. Mathematische Annalen, 357 (4), 1485-1508.

Scopus Eid


  • 2-s2.0-84887266385

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2493&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1484

Has Global Citation Frequency


Number Of Pages


  • 23

Start Page


  • 1485

End Page


  • 1508

Volume


  • 357

Issue


  • 4

Place Of Publication


  • Germany

Abstract


  • In this paper we study the functional W , which is the the sum of the Willmore energy, weighted surface area, and 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 nonnegative surface area weight 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.

Publication Date


  • 2013

Citation


  • McCoy, J. & Wheeler, G. E. (2013). A classification theorem for Helfrich surfaces. Mathematische Annalen, 357 (4), 1485-1508.

Scopus Eid


  • 2-s2.0-84887266385

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2493&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1484

Has Global Citation Frequency


Number Of Pages


  • 23

Start Page


  • 1485

End Page


  • 1508

Volume


  • 357

Issue


  • 4

Place Of Publication


  • Germany