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Morse index stability for Yang-Mills connections

Mario Gauvrit, Paul Laurain

Abstract

We prove stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of a bounded sequence of Yang-Mills connections is asymptotically bounded above by the sum of the Morse index and the nullity of the weak limit and the bubbles while the Morse indices are asymptotically bounded below by the sum of the Morse index of the weak limit and the bubbles.

Morse index stability for Yang-Mills connections

Abstract

We prove stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of a bounded sequence of Yang-Mills connections is asymptotically bounded above by the sum of the Morse index and the nullity of the weak limit and the bubbles while the Morse indices are asymptotically bounded below by the sum of the Morse index of the weak limit and the bubbles.
Paper Structure (14 sections, 37 theorems, 228 equations)

This paper contains 14 sections, 37 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Second variation : computation and finiteness of the Morse index
  3. Contribution of the necks to the second variation
  4. Positive contribution of the necks
  5. The need for sharp neck estimates
  6. Morse index semi-continuity
  7. Annexes
  8. weak connections and its notion of convergence
  9. Functional analysis results
  10. Hardy-Poincaré type inequalities in neck regions
  11. Gaffney inequality
  12. Diagonalization of quadratic forms
  13. Spectral theory results
  14. A diagonalisability condition

Key Result

Theorem 1.1

There exists $\epsilon_G >0$ and $C_G>0$ such that for all $A\in\mathfrak U_G(\mathrm{B}^4)$ satisfying there exists $g\in \mathrm{W}^{1,(4,\infty)}(\mathrm{B}^4,G)$ such that

Theorems & Definitions (73)

  • Theorem 1.1: theorem 1.3 UhlenbeckKarenK1982CwLb, theorem IV.1 rivière2015variations and theorem \ref{['extract']} of the appendix
  • Theorem 1.2: theorem VII.1 rivière2015variations
  • Theorem 1.3: theorem VII.3 rivière2015variations
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark
  • ...and 63 more