The Cartan–Hadamard conjecture and the Little Prince

Abstract

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