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