The Cartan-Hadamard conjecture and The Little Prince

Abstract

The generalized Cartan-Hadamard conjecture says that if $Ω$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le κ\le 0$, then the boundary of $Ω$ has the least possible boundary volume when $Ω$ is a round $n$-ball with constant curvature $K=κ$. The case $n=2$ and $κ=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $κ=0$, and a special case of the conjecture for $κ\textless{} 0$ and a version for $κ\textgreater{} 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $κ=0$. The generalization to $n=4$ and $κ\ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le κ$ to a weaker candle condition $Candle(κ)$ or $LCD(κ)$.We also find counterexamples to a naïve version of the Cartan-Hadamard conjecture: For every $\varepsilon \textgreater{} 0$, there is a Riemannian 3-ball $Ω$ 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.

v3: significant rewritting of some proofs, a mistake in the proof of the ball counter-example has been corrected