Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds

TL;DR

This paper derives an explicit formula relating total curvature of level sets to the isoperimetric inequality in Cartan-Hadamard manifolds, advancing understanding of geometric inequalities in nonpositively curved spaces.

Contribution

It introduces a new explicit formula for total curvature comparison, with applications to isoperimetric problems in nonpositive curvature manifolds.

Findings

Explicit curvature comparison formula derived

Applications to isoperimetric inequalities demonstrated

Enhanced understanding of geometric properties in Cartan-Hadamard spaces

Abstract

We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.