Relaxed Yang-Mills functional over 4-manifolds
type: Item
instance of: scholarly article
title: Relaxed Yang-Mills functional over 4-manifolds
zbMATH Open document ID: 0874.58008
DOI: 10.12775/TMNA.1995.043
author: Takeshi Isobe
published in: Topological Methods in Nonlinear Analysis
publication date: 1997-10-14T00:00:00Z
review text: Let \(M\) be a \(4\)-dimensional compact orientable Riemannian manifold and \({\mathcal P}_k\) the space of all principal \(SU(2)\)-bundles over \(M\) with second Chern number \(-k\). For a \(SU(2)\)-bundle \(P\) in \({\mathcal P}_k\), the second Chern class can be considered as a Radon measure on \(M\). Denote by \({\mathcal R}(M)\) the space of Radon measures on \(M\). Then, given a sequence of connections \(A_i\) with bounded Yang-Mills energy, the weak limit of the measure \(\text{ tr} (F_{A_i}\wedge F_{A_i})/8\pi^2\), if it exists, comprises a subspace in \({\mathcal R}(M)\), which is denoted by \({\mathcal P}^2_k\) in the author's notation. The author proves that any element in this space is written as \(\text{ tr} (F_{A_i}\wedge F_{A_i})/8\pi^2 + \sum d_j\delta_{a_j}\), where \(d_j\in {\mathbb{Z}}\) and \(\delta_{a_j}\) is the Dirac measure supported at \(a_j\), that the Yang-Mills functional can be extended over the space \({\mathcal P}^2_k\), and that any minimizing sequence relative to this extended functional has limit inside \({\mathcal P}^2_k\). A regularity theorem of the limit is given in certain cases.
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 929668
zbMATH Keywords: Yang-Mills functional
zbMATH Keywords: \(SU(2)\)-bundle
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: tmna.1995.043
OpenAlex ID: W2588128062
description: scientific article; zbMATH DE number 929668