Willmore energy

In differential geometry , the Willmore energy is a quantity that measures the bending energy of surfaces embedded in space. It is named after Thomas Willmore .

definition

For a smooth , embedded , compact , oriented surface with medium curvature , one defines the Willmore energy ${\displaystyle \Sigma \subset \mathbb {R} ^{3}}$ ${\displaystyle H:\Sigma \rightarrow \mathbb {R} }$

${\displaystyle W(\Sigma )=\int _{\Sigma }H^{2}dA}$.

motivation

Minimal surfaces in are, by definition, faces whose mean curvature vanishes . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle H\equiv 0}$

From the maximum principle it follows that there are no compact minimum surfaces without an edge. Instead, one looks for closed areas that minimize the Willmore energy. ${\displaystyle \mathbb {R} ^{3}}$

variant

Occasionally the Willmore energy gets through too

${\displaystyle \int _{\Sigma }(H^{2}-K)dA}$

defined with the Gaussian curvature . ${\displaystyle K:\Sigma \rightarrow \mathbb {R} }$

Because according to Gauss-Bonnet's theorem

${\displaystyle \int _{\Sigma }KdA=2\pi \chi (\Sigma )=4\pi (1-g)}$

applies, the two definitions only differ in one constant ( depending on the topology of the surface ). ${\displaystyle \Sigma }$

Spheres

A round sphere of any radius has Willmore energy . An elementary application of the inequality between arithmetic and geometric mean (together with the Gauss-Bonnet theorem ) shows that for every other sphere the Willmore energy is greater than . ${\displaystyle r}$ ${\displaystyle W(S^{2}(r))=4\pi }$ ${\displaystyle 4\pi }$

Tori

Clifford tori have Willmore energy . ${\displaystyle W(T)=2\pi ^{2}}$

Thomas Willmore hypothesized in 1965 that the inequality for every area is gender ${\displaystyle \geq 1}$

${\displaystyle W(\Sigma )\geq 2\pi ^{2}}$

applies. A proof of this conjecture was announced in February 2012 by Fernando Codá Marques and André Neves . Martin Schmidt already presented a proof of the Willmore conjecture in 2002, the completeness of which is, however, controversial in the professional world.

Immersions

Willmore energy can also be defined for immersions . Li and Yau have shown that for any non-embedded immersed surface the Willmore energy is at least . In particular, the minimum of Willmore energy among immersed spheres and tori is actually realized by embedded surfaces. ${\displaystyle f:\Sigma \rightarrow \mathbb {R} ^{3}}$ ${\displaystyle 8\pi }$

For immersed projective planes the Willmore energy is at least , the minimum is realized by the Bryant-Kusner parameterization of the Boy surface . ${\displaystyle 16\pi }$

Web links

• Yann Bernard: Autour des surfaces de Willmore

Individual evidence

1. ↑ The Willmore Conjecture after Marques and Neves
2. ↑ TJ Willmore: Note on embedded surfaces An. Sti. Univ. Al. I. Cuza Iasi, N. Ser., Sect. Ia 11B, 493-496 (1965)
3. ↑ Fernando Codá Marques, André Neves: Min-Max theory and the Willmore conjecture , arxiv : 1202.6036
4. ^ Martin U. Schmidt: A proof of the Willmore conjecture . In: arXiv . 2002. arxiv : math / 0203224 .