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 po ...
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 ...
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 geometry ...
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 tea ...
, may be traced back to work of
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
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 o ...
and
Viktor Zalgaller. In dimension 2 this fact had already been established in 1926 by
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
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 ...
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 ...
" by the probability theorist
Paul Lévy.
Weil's proof relies on
conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s and
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
, 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) transformat ...
, 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 and
Joel Spruck
Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University ...
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 PhD ...
, 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