mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of

Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...

for

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

s. The theory started with the efforts for generalizing

George David Birkhoff George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body ...

's method for the construction of simple closed

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

s on the sphere, to allow the construction of embedded

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 in arbitrary

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

s. It has played roles in the solutions to a number of

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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

s in

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

found by Almgren and Pitts themselves and also by other mathematicians, such as Mikhail Gromov,

Richard Schoen Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps. Earl ...

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

, Fernando Codá Marques, André Neves,

Ian Agol Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 a ...

, among others.

Description and basic concepts

The theory allows the construction of embedded minimal hypersurfaces through variational methods. In his PhD thesis, Almgren proved that the m-th

homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...

of the space of flat k-dimensional cycles on a closed

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

is isomorphic to the (m+k)-th dimensional homology group of M. This result is a generalization of the

Dold–Thom theorem In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common version of its proof consists of showing that the ...

, which can be thought of as the k=0 case of Almgren's theorem. Existence of non-trivial homotopy classes in the space of cycles suggests the possibility of constructing minimal submanifolds as saddle points of the volume function, as in

. In his subsequent work Almgren used these ideas to prove that for every k=1,...,n-1 a closed n-dimensional Riemannian manifold contains a stationary integral k-dimensional

varifold In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general alg ...

, a generalization of minimal submanifold that may have singularities. Allard showed that such generalized minimal submanifolds are regular on an open and dense subset. In the 1980s Almgren's student Jon Pitts greatly improved the regularity theory of minimal submanifolds obtained by Almgren in the case of codimension 1. He showed that when the dimension n of the manifold is between 3 and 6 the minimal hypersurface obtained using Almgren's min-max method is smooth. A key new idea in the proof was the notion of 1/j-almost minimizing varifolds.

and

Leon Simon Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of geometric analysis, ...

extended this result to higher dimensions. More specifically, they showed that every n-dimensional Riemannian manifold contains a closed minimal hypersurface constructed via min-max method that is smooth away from a closed set of dimension n-8. By considering higher parameter families of codimension 1 cycles one can find distinct minimal hypersurfaces. Such construction was used by Fernando Codá Marques and André Neves in their proof of the

Willmore conjecture In differential geometry, 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. A proof by Fernando Codá Marques and André Neves was ...

.Marques, Fernando & Neves, André. (2020). Applications of Min–Max Methods to Geometry. 10.1007/978-3-030-53725-8_2.

Description and basic concepts

See also

References

Further reading