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 $\partial\Omega$ 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 a counterexample to a na\"{\i}ve version of theCartan-Hadamard conjecture: We establish that for every $A, V >0$, there is a 3-ball with curvature $K \le -1$, volume $V$, and surfacearea $A$. 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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