The Cartan–Hadamard conjecture and the Little Prince

• Benoît R. Kloeckner ^[1] ; Greg Kuperberg ^[2]
  1. [1] Université Paris-Est – Créteil Val-de-Marne, Créteil
  2. [2] University of California at Davis
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 35, Nº 4, 2019, págs. 1195-1258
• Idioma: inglés
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• Resumen

  • 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≤κ≤0, then ∂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<0 and a version for κ>0.