differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the Willmore conjecture is a

lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

on the

Willmore energy In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is def ...

of a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

. It is named after the

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Tom Willmore, who conjectured it in 1965. A proof by

Fernando Codá Marques Fernando Codá dos Santos Cavalcanti Marques (born 8 October 1979) is a Brazilian mathematician working mainly in geometry, topology, partial differential equations and Morse theory. He is a professor at Princeton University. In 2012, together ...

and

André Neves André da Silva Graça Arroja Neves (born 1975, Lisbon) is a Portuguese mathematician and a professor at the University of Chicago. He joined the faculty of the University of Chicago in 2016. In 2012, jointly with Fernando Codá Marques, he sol ...

was announced in 2012 and published in 2014.

Willmore energy

Let ''v'' : ''M'' → R³ be a

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux * ''Immersion'' (album), the third album by Australian group Pendulum * ''Immersion'' (film), a 2021 ...

of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, orientable surface. Giving ''M'' the

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

induced by ''v'', let ''H'' : ''M'' → R be the

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

(the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

of the

principal curvature In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...

s ''κ''₁ and ''κ''₂ 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'') ≥ 4''π'', with equality

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

''M'' is an embedded round

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

Statement

Calculation of ''W''(''M'') for a few examples suggests that there should be a better bound than ''W''(''M'') ≥ 4''π'' for surfaces with

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

''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'') ≥ 2''π''². In 1982,

Peter Wai-Kwong Li Peter Wai-Kwong Li (born 18 April 1952) is a mathematician whose research interests include differential geometry and partial differential equations, in particular geometric analysis. After undergraduate work at California State University, Fresn ...

and

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...

proved the conjecture in the non-embedded case, showing that if

f:\Sigma\to S^3

is an immersion of a compact surface, which is ''not'' an embedding, then ''W''(''M'') is at least 8''π''. In 2012,

and

proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces.

Frank Morgan Francis Phillip Wuppermann (June 1, 1890 – September 18, 1949), known professionally as Frank Morgan, was an American character actor. He was best known for his appearances in films starting in the silent era in 1916, and then numerous soun ...

(2012)
Math Finds the Best Doughnut
, ''

The Huffington Post ''HuffPost'' (formerly ''The Huffington Post'' until 2017 and sometimes abbreviated ''HuffPo'') is an American progressive news website, with localized and international editions. The site offers news, satire, blogs, and original content, and ...

'' Martin Schmidt claimed a proof in 2002, 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 In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...

(by Langer & Singer).

References

{{reflist Conjectures that have been proved Surfaces Theorems in differential geometry de:Willmore-Energie