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 ...

, Yau's conjecture from 1982, is a

mathematical conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

which states that a closed Riemannian

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

has an infinite number of

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 ...

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

immersed

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s. It is named after

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 ...

. It was the first problem in the minimal submanifolds section in Yau's list of open problems. The conjecture has recently been claimed by Kei Irie,

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 solv ...

in the

generic Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...

case, and by Antoine Song in full generality.

References

Further reading