Hopf maps are unique Lipschitz constant minimizers

Among maps f: S^n –> S^m with nonzero Hopf invariant, we prove that Hopf maps uniquely minimize the Lipschitz constant. The proof uses elementary geometry and topology, and no analysis. We proved similar results about maps S^n –> S^n x S^n, vector fields on spheres, and a low dimension Stiefel projection. These positive results contrast nicely with several known counterexamples in calibration theory and harmonic map theory. This work is joint with H. Gluck and D. DeTurck.