Willmore conjecture

In differential geometry in mathematics the Willmore conjecture is a conjecture about the Willmore energy of a torus, named after the English mathematician Tom Willmore.^[1]

Willmore energy

Let v : M → R³ be a smooth immersion of a compact, orientable surface. 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

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.

The conjecture

Calculation of W(M) for a few examples suggests that there should be a better bound than W(M) ≥ 4π 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 every smooth immersed torus M in R³, W(M) ≥ 2π².

In 2012, Fernando Codá Marques and André Neves proved the conjecture using the min-max theory of minimal surfaces.^[2]

References