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Revista Matemática Iberoamericana

Volume 35, Issue 4, 2019, pp. 1195–1258
DOI: 10.4171/rmi/1082

Published online: 2019-07-08

The Cartan–Hadamard conjecture and the Little Prince

Benoît R. Kloeckner[1] and Greg Kuperberg[2]

(1) Université Paris-Est – Créteil Val-de-Marne, Créteil, France
(2) University of California at Davis, USA

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 $\partial\Omega$ 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 $\epsilon > 0$, there is a Riemannian $\Omega \cong B^3$ with $(1-\epsilon)$-pinched negative curvature, and with $|\partial\Omega|$ bounded by a function of $\epsilon$ and $|\Omega|$ arbitrarily large.

We begin with a pointwise isoperimetric problem called "the problem of the Little Prince". Its proof becomes part of the more general method.

Keywords: Isoperimetric inequality, Cartan–Hadamard manifolds, Riemannian geometry, optimal transportation, linear programming, upper curvature bounds

Kloeckner Benoît, Kuperberg Greg: The Cartan–Hadamard conjecture and the Little Prince. Rev. Mat. Iberoam. 35 (2019), 1195-1258. doi: 10.4171/rmi/1082