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Article Dans Une Revue Revista Matemática Iberoamericana Année : 2019

The Cartan-Hadamard conjecture and The Little Prince

Résumé

The generalized Cartan-Hadamard conjecture says that if $\Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le \kappa \le 0$, then the boundary of $\Omega$ has the least possible boundary volume when $\Omega$ is a round $n$-ball with constant curvature $K=\kappa$. The case $n=2$ and $\kappa=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $\kappa=0$, and a special case of the conjecture for $\kappa < 0$ and a version for $\kappa > 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $\kappa=0$. The generalization to $n=4$ and $\kappa \ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le \kappa$ to a weaker candle condition $Candle(\kappa)$ or $LCD(\kappa)$. We also find counterexamples to a naïve version of the Cartan-Hadamard conjecture: For every $\varepsilon > 0$, there is a Riemannian 3-ball $\Omega$ with $(1-\varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $\varepsilon$ and with arbitrarily large volume. We begin with a pointwise isoperimetric problem called "the problem of the Little Prince.'' Its proof becomes part of the more general method.
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Dates et versions

hal-00800027 , version 1 (13-03-2013)
hal-00800027 , version 2 (24-11-2014)
hal-00800027 , version 3 (13-02-2017)

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Benoît Kloeckner, Greg Kuperberg. The Cartan-Hadamard conjecture and The Little Prince. Revista Matemática Iberoamericana, 2019, 35 (4), pp.1195-1258. ⟨10.4171/RMI/1082⟩. ⟨hal-00800027v3⟩
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