Cartan–Hadamard conjecture

In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, 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, or ripples on a petunia 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,^[1] and a few years later by Misha Gromov,^[2][3]Yuri Burago and Viktor Zalgaller.^[4][5] In dimension 2 this fact had already been established in 1926 by André Weil^[6] and rediscovered in 1933 by Beckenbach and Rado.^[7] In dimensions 3 and 4 the conjecture was proved by Bruce Kleiner^[8] in 1992, and Chris Croke^[9] in 1984 respectively.

According to Marcel Berger,^[10] 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" 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, while Kleiner adopts a variational approach which reduces the problem to an estimate for total curvature. Mohammad Ghomi and Joel Spruck have shown that Kleiner's approach will work in all dimensions where the total curvature inequality holds.^[11]

Generalized form

The conjecture has a more general form, sometimes called the "generalized Cartan–Hadamard conjecture"^[12] 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,^[13] and dimension 3 by Kleiner.^[14] The generalized conjecture also holds for regions of small volume in all dimensions, as proved by Frank Morgan and David Johnson.^[15]

Applications

Immediate applications of the conjecture include extensions of the Sobolev inequality and Rayleigh–Faber–Krahn inequality to spaces of nonpositive curvature.

References

1. Aubin, Thierry (1976). "Problèmes isopérimétriques et espaces de Sobolev". Journal of Differential Geometry. 11 (4): 573–598. doi:10.4310/jdg/1214433725. ISSN 0022-040X.
2. Gromov, Mikhael, 1943- (1999). Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser. ISBN 0817638989. OCLC 37201427.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
3. Gromov, Mikhael (1981). Structures métriques pour les variétés riemanniennes (in French). CEDIC/Fernand Nathan. ISBN 9782712407148.
4. Burago, Yuri; Zalgaller, Viktor (1980). Geometricheskie neravenstva. "Nauka, " Leningradskoe otd-nie. OCLC 610467367.
5. Burago, Yuri; Zalgaller, Viktor (1988). Geometric Inequalities. doi:10.1007/978-3-662-07441-1. ISBN 978-3-642-05724-3.
6. Weil, M. André; Hadamard, M. (1979), "Sur les surfaces à courbure négative", Œuvres Scientifiques Collected Papers, Springer New York, pp. 1–2, doi:10.1007/978-1-4757-1705-1_1, ISBN 9781475717068
7. Beckenbach, E. F.; Rado, T. (1933). "Subharmonic Functions and Surfaces of Negative Curvature". Transactions of the American Mathematical Society. 35 (3): 662. doi:10.2307/1989854. ISSN 0002-9947. JSTOR 1989854.
8. Kleiner, Bruce (1992). "An isoperimetric comparison theorem". Inventiones Mathematicae. 108 (1): 37–47. Bibcode:1992InMat.108...37K. doi:10.1007/bf02100598. ISSN 0020-9910. S2CID 16836013.
9. Croke, Christopher B. (1984). "A sharp four dimensional isoperimetric inequality". Commentarii Mathematici Helvetici. 59 (1): 187–192. doi:10.1007/bf02566344. ISSN 0010-2571. S2CID 120138158.
10. Berger, Marcel. (2013). A Panoramic View of Riemannian Geometry. Springer Berlin. ISBN 978-3-642-62121-5. OCLC 864568506.
11. Ghomi, Mohammad; Spruck, Joel (2022-01-04). "Total Curvature and the Isoperimetric Inequality in Cartan–Hadamard Manifolds". The Journal of Geometric Analysis. 32 (2): 50. arXiv:1908.09814. doi:10.1007/s12220-021-00801-2. ISSN 1559-002X. S2CID 255558870.
12. Kloeckner, Benoît; Kuperberg, Greg (2019-07-08). "The Cartan–Hadamard conjecture and the Little Prince". Revista Matemática Iberoamericana. 35 (4): 1195–1258. arXiv:1303.3115. doi:10.4171/rmi/1082. ISSN 0213-2230. S2CID 119165853.
13. Bol, G. Isoperimetrische Ungleichungen für Bereiche auf Flächen. OCLC 946388942.
14. Kleiner, Bruce (1992). "An isoperimetric comparison theorem". Inventiones Mathematicae. 108 (1): 37–47. Bibcode:1992InMat.108...37K. doi:10.1007/bf02100598. ISSN 0020-9910. S2CID 16836013.
15. Morgan, Frank; Johnson, David L. (2000). "Some sharp isoperimetric theorems for Riemannian manifolds". Indiana University Mathematics Journal. 49 (3): 0. doi:10.1512/iumj.2000.49.1929. ISSN 0022-2518.