Willmore conjecture

In mathematics — specifically, in differential geometry — the Willmore conjecture is a conjecture about the Willmore energy of a torus. The conjecture is named after the English mathematician Tom Willmore.

tatement of the conjecture

Let "v" : "M" → R³ be a smooth immersion of a compact, orientable surface (of dimension two). Giving "M" the Riemannian metric induced by "v", let "H" : "M" → R be the mean curvature (the arithmetic mean of the principal curvatures "κ"₁ and "κ"₂ at each point). In this notation, the Willmore energy "W"("M") of "M" is given by

:$W(M)\; =\; int\_\{M\}\; H^\{2\}.$

It is not hard to prove that the Willmore energy satisfies "W"("M") ≥ 4"π", with equality if and only if "M" is an embedded round sphere. Calculation of "W"("M") for a few examples suggests that there should be a better bound for surfaces with genus "g"("M") > 0. In particular, calculation of "W"("M") for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name: for any smooth immersed torus "M" in R³, "W"("M") ≥ 2"π"².

References

* cite journal
last = Topping
first = Peter M.
title = Towards the Willmore conjecture
journal = Calc. Var. Partial Differential Equations
volume = 11
year = 2000
issue = 4
pages = 361–393
issn = 0944-2669
doi = 10.1007/s005260000042 MathSciNet|id=1808127
* cite journal
last = Willmore
first = Thomas J.
title = Note on embedded surfaces
journal = An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.)
volume = 11B
year = 1965
pages = 493–496 MathSciNet|id=0202066