Vol. 7, No. 8, 2014

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Global gauges and global extensions in optimal spaces

Mircea Petrache and Tristan Rivière

Vol. 7 (2014), No. 8, 1851–1899
Abstract

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

Keywords
nonlinear extension, nonlinear Sobolev space, global gauge, conformally invariant problem, Yang–Mills, Lorentz spaces, Hopf lift
Mathematical Subject Classification 2010
Primary: 28A51, 46E35
Secondary: 70S15, 58J05