The paper proves in all dimensions that the total curvature inequality implies the isoperimetric inequality in Cartan-Hadamard manifolds, and establishes a new comparison formula for total curvature of level sets.
Cut and conjugate points of the exponential map, with applications
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abstract
The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and to derive applications to the Ambrose conjecture and the Hamilton-Jacobi equations.
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Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds
The paper proves in all dimensions that the total curvature inequality implies the isoperimetric inequality in Cartan-Hadamard manifolds, and establishes a new comparison formula for total curvature of level sets.