The min-max width of spheres associated to the distance function

2025/03/16 by Montezuma, Rafael, Ribeiro, Idalina
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.12477

Abstract

What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.

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