Willmore conjecture

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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[edit]

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

 W(M) = \int_M H^2 \, dA.

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[edit]

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[edit]

  1. ^ Willmore, Thomas J. (1965). "Note on embedded surfaces". An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B: 493–496. 
  2. ^ Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. arXiv:1202.6036.