The Cartan–Hadamard conjecture and the Little Prince

  • Benoît R. Kloeckner

    Université Paris-Est – Créteil Val-de-Marne, Créteil, France
  • Greg Kuperberg

    University of California at Davis, USA
The Cartan–Hadamard conjecture and the Little Prince cover
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Abstract

The generalized Cartan–Hadamard conjecture says that if is a domain with fixed volume in a complete, simply connected Riemannian -manifold with sectional curvature , then has the least possible boundary volume when is a round -ball with constant curvature . The case and is an old result of Weil. We give a unified proof of this conjecture in dimensions and when , and a special case of the conjecture for and a version for . Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for and . The generalization to and is a new result. As Croke implicitly did, we relax the curvature condition to a weaker candle condition Candle} or LCD}.

We also find counterexamples to a naïve version of the Cartan–Hadamard conjecture: For every , there is a Riemannian with -pinched negative curvature, and with bounded by a function of and 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.

Cite this article

Benoît R. Kloeckner, Greg Kuperberg, The Cartan–Hadamard conjecture and the Little Prince. Rev. Mat. Iberoam. 35 (2019), no. 4, pp. 1195–1258

DOI 10.4171/RMI/1082