Willmore Surface
Willmore Surface is a topological surface, a critical point of the Willmore energy, given by a parametrisation, immersed.
Open Willmore Surface in the interactive viewer →
aperiodic def-parametric immersed implemented topological willmore
Properties
- Family
- topological
- Given by
- a parametrisation
- Curvature
- critical point of the Willmore energy
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
No chart is stored: produced by gradient-flow relaxation of the Willmore energy from a seed torus (make_willmore_torus, hundreds of iterations); the result has no parametrization at all. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- T. J. Willmore, "Note on embedded surfaces", An. Sti. Univ. "Al.
- I. Cuza" Iasi Sect. I a Mat. 11B (1965), 493-496.
- F. C. Marques and A. Neves, "Min-max theory and the Willmore conjecture", Annals of Mathematics 179(2) (2014), 683-782.
- W. Helfrich, "Elastic properties of lipid bilayers: theory and possible experiments", Z. Naturforsch. C 28 (1973), 693-703.
- U. Seifert, K. Berndl, R. Lipowsky, "Shape transformations of vesicles", Physical Review A 44 (1991), 1182-1202 -- the discocyte/prolate/stomatocyte phase diagram at reduced volume < 1.
- C. Zhu, C. T. Lee, P. Rangamani, "Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry", Biophysical Reports 2(3), 100062 (2022).