The Cartan-Hadamard conjecture and the Little Prince

The generalized Cartan-Hadamard conjecture says that if Omega is a domain with fixed volume in a complete, simply connected Riemannian n-manifold M with sectional curvature K <= kappa <= 0, then partial derivative Omega has the least possible boundary volume when Omega is a round n: ball with constant. curvature K = kappa. The case n = 2 and kappa = 0 is an old result of Weil. We give a unified proof of this conjecture in dimensions n = 2 and n = 4 when kappa = 0, and a special case of the conjecture for kappa < 0 and a version for kappa > 0. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for n = 4 and kappa = 0. The generalization to n = 4 and kappa not equal 0 is a new result. As Croke implicitly did, we relax the curvature condition K <= kappa to a weaker candle condition Candle(kappa) or LCD(kappa). We also find counterexamples to a naive version of the Cartan-Hadamard conjecture: For every c > 0, there is a Riemannian Omega similar to B-3 with (1 - c)-pinched negative curvature, and with vertical bar partial derivative Omega vertical bar bounded by a function of c and vertical bar Omega vertical bar 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.