Lipschitz minimality of Hopf fibrations and Hopf vector fields

TL;DR

This paper proves that Hopf fibrations and associated vector fields on spheres uniquely minimize Lipschitz constants within their homotopy classes, establishing their optimality in a mathematical sense.

Contribution

It demonstrates the Lipschitz minimality and uniqueness of Hopf fibrations and vector fields on spheres, a problem previously only partially understood.

Findings

Hopf fibrations are the unique Lipschitz minimizers in their homotopy class.

Hopf vector fields tangent to fibers are also Lipschitz minimizers.

The results clarify the optimality of Hopf structures in geometric analysis.

Abstract

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.