2013/03/13 by Kloeckner, Benoît, Kuperberg, Greg · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1303.3115
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 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≠ 0 is a new result. As Croke implicitly did, we relax the curvature condition K ≤ 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 ε \textgreater 0, there is a Riemannian 3-ball Ω with (1-ε)-pinched negative curvature, and with boundary volume bounded by a function of ε 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.