The Cartan-Hadamard conjecture and The Little Prince

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 $kappa textless{} 0$ and a version for $kappa textgreater{} 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Crokes proof for $n=4$ and $kappa=0$. The generalization to $n=4$ and $kappa e 0$ 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 naive version of the Cartan-Hadamard conjecture: For every $varepsilon textgreater{} 0$, 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.