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The stability of Yang-Mills connections on $δ$-pinched manifolds

Xiaoli Han, Yang Wen

TL;DR

This work addresses rigidity of weakly stable Yang-Mills connections on compact, δ-pinched manifolds. It introduces a dimension-dependent pinching constant δ(n) by constructing specialized variation fields $V_y$ and applying the Bochner–Weitzenböck framework to estimate the second variation along $i_{V_y}R^\nabla$, reducing the problem to the sign of a radial integral involving a function $\Phi_\delta$ with weight $v_\delta$. By optimizing over δ via volume comparison and injectivity radius considerations, the authors prove that in the δ-pinched regime, all weakly stable Yang-Mills connections are flat, and they provide computed δ(n) values for dimensions $n=5$–$20$; the approach also yields a generalized δ(n,R) for varying injectivity radius. The results extend rigidity phenomena for YM fields in pinched geometries and offer concrete geometric constants for classification of YM connections on such manifolds.

Abstract

In this article, we establish pinching conditions under which all weakly stable Yang-Mills connections on compact manifolds are flat. As a corollary, we provide a dimension-dependent constant $δ(n)$ and prove that there exist no non-flat weakly stable Yang-Mills connections on $δ(n)$-pinched compact simply-connected Riemannian manifolds.

The stability of Yang-Mills connections on $δ$-pinched manifolds

TL;DR

This work addresses rigidity of weakly stable Yang-Mills connections on compact, δ-pinched manifolds. It introduces a dimension-dependent pinching constant δ(n) by constructing specialized variation fields and applying the Bochner–Weitzenböck framework to estimate the second variation along , reducing the problem to the sign of a radial integral involving a function with weight . By optimizing over δ via volume comparison and injectivity radius considerations, the authors prove that in the δ-pinched regime, all weakly stable Yang-Mills connections are flat, and they provide computed δ(n) values for dimensions –; the approach also yields a generalized δ(n,R) for varying injectivity radius. The results extend rigidity phenomena for YM fields in pinched geometries and offer concrete geometric constants for classification of YM connections on such manifolds.

Abstract

In this article, we establish pinching conditions under which all weakly stable Yang-Mills connections on compact manifolds are flat. As a corollary, we provide a dimension-dependent constant and prove that there exist no non-flat weakly stable Yang-Mills connections on -pinched compact simply-connected Riemannian manifolds.
Paper Structure (8 sections, 11 theorems, 78 equations, 1 table)

This paper contains 8 sections, 11 theorems, 78 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The connections and curvatures on vector bundles
  4. Estimate for the second variation of Yang-Mills functional
  5. Setting and Notation
  6. First-order derivative estimate for the vector field $V_y$
  7. The second variation formula
  8. Proof of the main result

Key Result

Theorem 1.1

Let $M$ be a compact $n$-dimensional Riemannian manifold with $n\ge5$ and $\operatorname{inj}(M) \geq c_n\pi$ for some $c_n \in (0,1]$. There exists $\delta = \delta(n, \operatorname{inj}(M)) > 0$ such that if the sectional curvature satisfies $\delta < K \leq 1$, then every weakly stable Yang-Mills

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.1
  • Proposition 4.1: BM Section 6
  • ...and 6 more