The distance between homotopy classes of Sobolev maps on spheres

Rupert L. Frank; Paata Ivanisvili

The distance between homotopy classes of Sobolev maps on spheres

Analysis of PDEs 2026-06-28 v1

Authors: Rupert L. Frank , Paata Ivanisvili

Abstract

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees. In the case of the 2-sphere this resolves an open problem by Brezis.

Keywords

Sobolev maps homotopy classes Brouwer degree sphere self-maps critical Sobolev space

Cite

@article{arxiv.2606.29382,
  title  = {The distance between homotopy classes of Sobolev maps on spheres},
  author = {Rupert L. Frank and Paata Ivanisvili},
  journal= {arXiv preprint arXiv:2606.29382},
  year   = {2026}
}