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WILMORE'S TORUS

Links:
Wikipedia article on the Willmore conjecture www.chem.ucla.edu/~michalet/papers/larecherche/f-tores.html |

Willmore's torus is the torus
for which the ratio of the major and minor axes is equal to .

The *energy of curvature* of a surface *S*,
or *total mean curvature*, beeing defined by the integral
where
is the mean curvature, the Willmore's torus is the torus with minimal energy.
Thus, for a torus of major axe
*a* and minor axe *b, *the energy
*E*
is , that
is minimal when ,
with the value .

Willmore conjectured in 1965 that is the minimal value of the energy of all orientable surface of genus 1 (conjecture proved in 2014). The surfaces of genus 1 reaching this minimum value are Willmore's torus and its inverses (thus particular Dupin cyclids).

This was verified through experimentation by the shapes assumed by liposomes.

Note that the *total absolute Gaussian curvature*,
defined by
where
is for
each torus. Sea also the tight surfaces.

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© Robert FERRÉOL
2019