Vol. 7, No. 8, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 ϕ : Sn Sn to functions u : Bn+1 Sn for n = 2,3. We assume ϕ belongs to the critical space W1,n and we construct a W1,(n+1,)-controlled extension u. The Lorentz–Sobolev space W1,(n+1,) is optimal for such controlled extension. Then we use these results to construct global controlled gauges for L4-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.

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