Global gauges and global extensions in optimal spaces

TL;DR

This paper develops optimal Lorentz-Sobolev space extensions for functions into spheres and applies these results to construct global gauges for certain connections in four dimensions, extending local Sobolev control to a global setting.

Contribution

It introduces a new optimal Lorentz-Sobolev space extension method for functions into spheres and applies it to global gauge constructions for SU(2)-connections.

Findings

Constructed W^{1,(n+1, Infty)}-controlled extensions for functions in W^{1,n}

Established global gauges for L^4-connections in 4D

Extended local Sobolev control of connections to a global framework

Abstract

We consider the problem of extending functions \phi:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume \phi to belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such results to construct global controlled gauges for L^4-connections over trivial SU(2)-bundles in 4 dimensions. This result is a global version of the local Sobolev control of connections obtained by K. Uhlenbeck.