In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

and

Geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...

which states that the classical

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

may be generalized to spaces of nonpositive

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...

and

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...

, may be traced back to work of André Weil in 1926. Informally, the conjecture states that negative curvature allows regions with a given perimeter to hold more volume. This phenomenon manifests itself in nature through corrugations on

coral reefs A coral reef is an underwater ecosystem characterized by reef-building corals. Reefs are formed of colonies of coral polyps held together by calcium carbonate. Most coral reefs are built from stony corals, whose polyps cluster in groups. Co ...

, or ripples on a

petunia ''Petunia'' is genus of 20 species of flowering plants of South American origin. The popular flower of the same name derived its epithet from the French, which took the word ''petun'', meaning "tobacco," from a Tupi–Guarani language. A tende ...

flower, which form some of the simplest examples of non-positively curved spaces.

History

The conjecture, in all dimensions, was first stated explicitly in 1976 by

Thierry Aubin Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contrib ...

, and a few years later by Misha Gromov,

Yuri Burago Yuri Dmitrievich Burago (russian: Ю́рий Дми́триевич Бура́го) (born 1936) is a Russian mathematician. He works in differential and convex geometry. Education and career Burago studied at Leningrad University, where he ...

and

Viktor Zalgaller Victor (Viktor) Abramovich Zalgaller ( he, ויקטור אבּרמוביץ' זלגלר; russian: Виктор Абрамович Залгаллер; 25 December 1920 – 2 October 2020) was a Russian-Israeli mathematician in the fields of ge ...

. In dimension 2 this fact had already been established in 1926 by André Weil and rediscovered in 1933 by Beckenbach and Rado. In dimensions 3 and 4 the conjecture was proved by

Bruce Kleiner Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory. He received his Ph.D. in 1990 from the University of California, Berkeley. His advisor was Wu-Yi Hsiang. Kleiner is a p ...

in 1992, and Chris Croke in 1984 respectively. According to

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Las ...

, Weil, who was a student of Hadamard at the time, was prompted to work on this problem due to "a question asked during or after a Hadamard seminar at the

Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris n ...

" by the probability theorist Paul Lévy. Weil's proof relies on conformal maps and harmonic analysis, Croke's proof is based on an inequality of Santaló in

integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformati ...

, while Kleiner adopts a variational approach which reduces the problem to an estimate for

total curvature In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve i ...

Mohammad Ghomi Muhammad ( ar, مُحَمَّد; 570 – 8 June 632 CE) was an Arab religious, social, and political leader and the founder of Islam. According to Islamic doctrine, he was a prophet divinely inspired to preach and confirm the monoth ...

and Joel Spruck have shown that Kleiner's approach will work in all dimensions where the total curvature inequality holds.

Generalized form

The conjecture has a more general form, sometimes called the "generalized Cartan–Hadamard conjecture" which states that if the curvature of the ambient Cartan–Hadamard manifold M is bounded above by a nonpositive constant k, then the least perimeter enclosures in M, for any given volume, cannot have smaller perimeter than a sphere enclosing the same volume in the model space of constant curvature k. The generalized conjecture has been established only in dimension 2 by

Gerrit Bol Gerrit Bol (May 29, 1906 in Amsterdam – February 21, 1989 in Freiburg) was a Dutch mathematician who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points. Life Bol earned his Ph ...

, and dimension 3 by Kleiner. The generalized conjecture also holds for regions of small volume in all dimensions, as proved by

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 sound ...

and David Johnson.

Applications

Immediate applications of the conjecture include extensions of the

Sobolev inequality In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...

and

Rayleigh–Faber–Krahn inequality In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eig ...

to spaces of nonpositive curvature.

References

{{DEFAULTSORT:Cartan-Hadamard conjecture Riemannian geometry Measure theory Conjectures