Global gauges and global extensions in optimal spaces

Petrache, Mircea; Rivière, Tristan

doi:10.2140/apde.2014.7.1851

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Global gauges and global extensions in optimal spaces

Mircea Petrache and Tristan Rivière

Vol. 7 (2014), No. 8, 1851–1899

DOI: 10.2140/apde.2014.7.1851

Abstract

We consider the problem of extending functions $ϕ : S^{n} \to S^{n}$ to functions $u : B^{n + 1} \to S^{n}$ for $n = 2, 3$ . We assume $ϕ$ belongs 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 these results to construct global controlled gauges for $L^{4}$ -connections over trivial $S U (2)$ -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

Milestones

Received: 16 January 2014

Revised: 5 July 2014

Accepted: 11 August 2014

Published: 5 February 2015

Authors

Mircea Petrache
Laboratoire Jacques-Louis Lions Universitè Pierre et Marie Curie (Paris 6) place Jussieu, 4 75005 Paris France

Tristan Rivière
Department of Mathematics ETH Zürich CH-8092 Zürich Switzerland

Vol. 7, No. 8, 2014