Resumen de The Cartan–Hadamard conjecture and the Little Prince

Benoît R. Kloeckner, Greg Kuperberg

• 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.