Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 8 (2014), 1851-1899.

Global gauges and global extensions in optimal spaces

Mircea Petrache and Tristan Rivière

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We consider the problem of extending functions ϕ:SnSn to functions u:Bn+1Sn 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.

Article information

Anal. PDE, Volume 7, Number 8 (2014), 1851-1899.

Received: 16 January 2014
Revised: 5 July 2014
Accepted: 11 August 2014
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 28A51: Lifting theory [See also 46G15] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 70S15: Yang-Mills and other gauge theories 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]

nonlinear extension nonlinear Sobolev space global gauge conformally invariant problem Yang–Mills Lorentz spaces Hopf lift


Petrache, Mircea; Rivière, Tristan. Global gauges and global extensions in optimal spaces. Anal. PDE 7 (2014), no. 8, 1851--1899. doi:10.2140/apde.2014.7.1851.

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