Willmore conjecture
In differential geometry, an area of mathematics, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965.[1] A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014.[2][3]
Willmore energy
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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 Almgren–Pitts min-max theory of minimal surfaces.[2][3] Martin Schmidt had already announced a proof in 2002,[4] but it was not accepted for publication in any peer-reviewed mathematical journal.
References
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- ↑ 2.0 2.1 Frank Morgan (2012) "Math Finds the Best Doughnut", The Huffington Post
- ↑ 3.0 3.1 Lua error in package.lua at line 80: module 'strict' not found.
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