It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only ifM is an embedded round sphere.
Statement
Calculation of W(M) for a few examples suggests that there should be a better bound than W(M) ≥ 4π for surfaces with genusg(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 1982, Peter Wai-Kwong Li and Shing-Tung Yau proved the conjecture in the non-embedded case, showing that if is an immersion of a compact surface, which is not an embedding, then W(M) is at least 8π.[4]
In 2012, Fernando Codá Marques and André Neves proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces.[3][1] Martin Schmidt claimed a proof in 2002,[5] but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it). Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as tube tori (by Willmore himself), and for tori of revolution (by Langer & Singer).[6]
^Willmore, Thomas J. (1965). "Note on embedded surfaces". Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi, Secţiunea I a Matematică. 11B: 493–496. MR0202066.
^Li, Peter; Yau, Shing Tung (1982). "A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces". Inventiones Mathematicae. 69 (2): 269–291. doi:10.1007/BF01399507. MR0674407.
^Schmidt, Martin U. (2002). "A proof of the Willmore conjecture". arXiv:math/0203224.
^Langer, Joel; Singer, David (1984). "Curves in the hyperbolic plane and mean curvature of tori in 3-space". The Bulletin of the London Mathematical Society. 16 (5): 531–534. doi:10.1112/blms/16.5.531. MR0751827.