Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topology and approximation of weak $G$-bundles in the supercritical dimensions

Swarnendu Sil

Abstract

For analyzing stationary Yang-Mills connections in higher dimensions, one has to work with Morrey-Sobolev bundles and connections. The transition maps for a Morrey-Sobolev principal $G$-bundles are not continuous and thus the usual notion of topology does not make sense. In this work, we develop the notion of a topological isomorphism class for a bundle-connection pair $\left( P, A\right)$ and use these notions to derive several approximability results for bundles and connections in the Morrey-Sobolev setting. Our proofs follow a connection-oriented approach and also highlight the fact that in the low regularity regime, the regularity of the bundle and connection are intertwined. Our results parallels the theory of the topological degree and approximation results for manifold-valued $\mathrm{VMO}$ maps.

Topology and approximation of weak $G$-bundles in the supercritical dimensions

Abstract

For analyzing stationary Yang-Mills connections in higher dimensions, one has to work with Morrey-Sobolev bundles and connections. The transition maps for a Morrey-Sobolev principal -bundles are not continuous and thus the usual notion of topology does not make sense. In this work, we develop the notion of a topological isomorphism class for a bundle-connection pair and use these notions to derive several approximability results for bundles and connections in the Morrey-Sobolev setting. Our proofs follow a connection-oriented approach and also highlight the fact that in the low regularity regime, the regularity of the bundle and connection are intertwined. Our results parallels the theory of the topological degree and approximation results for manifold-valued maps.
Paper Structure (39 sections, 33 theorems, 238 equations)

This paper contains 39 sections, 33 theorems, 238 equations.

Table of Contents

  1. Introduction and main results
  2. Bundles and connections with bounds on $m$-Yang-Mills energy
  3. Topology without continuity
  4. Main results
  5. Coulomb gauges and continuity of the Coulomb bundles
  6. Organization of the article
  7. Notations and preliminaries
  8. Function spaces
  9. Morrey spaces and Vanishing Morrey spaces
  10. Morrey-Sobolev and Vanishing Morrey-Sobolev spaces
  11. Sobolev embedding for Morrey-Sobolev spaces
  12. Smooth principal $G$ bundles with connections
  13. Bundles as cocycles
  14. Good covers and refinements
  15. Connection, gauges and curvature
  16. ...and 24 more sections

Key Result

Theorem 1

Let $P$ be a principal $G$-bundle over a $n$-dimensional closed manifold $M^{n},$ such that the bundle transition functions are in the vanishing Morrey-Sobolev class $\mathsf{W}^{2}\mathrm{VL}^{2m,n-2m}$ and let $A$ be a connection on $P$ such that the local connection forms are in vanishing Morrey-

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Proposition 8: Hölder inequality in Morrey spaces
  • Proposition 9
  • Proposition 10
  • ...and 52 more