Willmore conjecture
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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.
[edit] Statement 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 κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by
| W(M) = | ∫ | H2. |
| M |
It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, 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) ≥ 2π2.
