Lipschitz minimality of Hopf fibrations and Hopf vector fields

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. Introduction and statement of results. The Hopf fibration S1 ⊂ S3 → S2 of a round 3-sphere by parallel great circles was introduced by Heinz Hopf [1931]. It provided the first example of a homotopically nontrivial map from one sphere to another of lower dimension, spurring the develop-ment of both homotopy theory and fibre spaces in their infancy. Although Hopf firs