A Note on the Geometry of Positively-Curved Riemannian Manifolds

TL;DR

This paper presents a comparison theorem for the waist of positively curved Riemannian manifolds, partially addressing Gromov's conjecture by combining classical volume comparison with geometric measure theory techniques.

Contribution

It introduces a new comparison theorem for the waist of positively curved manifolds, advancing understanding of their geometric structure.

Findings

Established a partial positive answer to Gromov's conjecture.

Developed a novel combination of volume comparison and Almgren-Pitts Min-Max theory.

Provided new insights into the geometry of positively curved Riemannian manifolds.

Abstract

In this paper I present a comparison theorem for the waist of Riemannian manifolds with positive sectional curvature. The main theorem of this paper gives a partial positive answer to a conjecture formulated by M.Gromov in [8]. The content of this paper combines two aspects: classical volume comparison theorems of Riemannian geometry, and geometric measure theoretic ideas stemming from Almgren-Pitts Min-Max theory