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Gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality

Matheus Vieira

Abstract

In this paper we prove gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to many examples of manifolds. We also prove a uniqueness theorem for the basic instanton.

Gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality

Abstract

In this paper we prove gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to many examples of manifolds. We also prove a uniqueness theorem for the basic instanton.

Paper Structure

This paper contains 8 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Weighted Poincaré inequalities
  3. Proofs
  4. Bochner formula
  5. Proof of Theorem \ref{['thm:vol']}
  6. Proof of Theorem \ref{['thm:int']}
  7. Proof of Corollary \ref{['cor:app']}
  8. Proof of Theorem \ref{['thm:adhm']}

Key Result

Theorem 1

Suppose a complete four-dimensional Riemannian manifold $X$, with scalar curvature $S$ and Weyl curvature $W$, satisfies the weighted Poincaré inequality (eq:wpi) with a weight function $q$ and has volume growth $vol\left(B_{R}\right)=O\left(R^{p}\right)$ for some constant $p>0$. Given a Yang-Mills Then either inequality (eq:volineq) is an equality (everywhere) or the self-dual curvature $F^{+}$

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 1 more