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The stability for F-Yang-Mills functional on CP^n

Yang Wen

TL;DR

The paper addresses stability of the $F$-Yang-Mills functional on complex projective space $\mathbb{C}P^n$ and characterizes when weakly stable connections must be flat or have rigid curvature form. It develops a second-variation framework for $\mathscr{A}_F(\nabla)=\int F(\tfrac12|R^\nabla|^2)$, and analyzes it via Killing-vector-field variations on $\mathbb{C}P^n$, leveraging the Bochner-Weitzenböck identity. The main results show that a negativity condition on $F''$ and $F'$ forces stability to imply trivial curvature, while equality yields a precise curvature structure $R^\nabla(X,Y)=g(X,JY)\,\sigma$, along with a gap theorem bounding $\|R^\nabla\|_{L^\infty}$ to force flatness. Together, these findings clarify stability and curvature constraints for generalized Yang–Mills functionals on symmetric spaces, with explicit corollaries for power-type $F$ on low-dimensional projective spaces.

Abstract

In this paper, we study the critical points of $F$-Yang-Mills functional on $\mathbb{C}P^n$, which are called $F$-Yang-Mills connections. We prove that if $(2+\frac4n)F''(x)x+(n+1)F'(x)<0$, then the weakly stable $F$-Yang-Mills connection on $\mathbb{C}P^n$ must be flat. Moreover, if $(2+\frac4n)F''(x)x+(n+1)F'(x)=0$, we obtain the structure of curvatures corresponding to weakly stable connections. We also show a gap theorem for $F$-Yang-Mills connections on $\mathbb{C}P^n$.

The stability for F-Yang-Mills functional on CP^n

TL;DR

The paper addresses stability of the -Yang-Mills functional on complex projective space and characterizes when weakly stable connections must be flat or have rigid curvature form. It develops a second-variation framework for , and analyzes it via Killing-vector-field variations on , leveraging the Bochner-Weitzenböck identity. The main results show that a negativity condition on and forces stability to imply trivial curvature, while equality yields a precise curvature structure , along with a gap theorem bounding to force flatness. Together, these findings clarify stability and curvature constraints for generalized Yang–Mills functionals on symmetric spaces, with explicit corollaries for power-type on low-dimensional projective spaces.

Abstract

In this paper, we study the critical points of -Yang-Mills functional on , which are called -Yang-Mills connections. We prove that if , then the weakly stable -Yang-Mills connection on must be flat. Moreover, if , we obtain the structure of curvatures corresponding to weakly stable connections. We also show a gap theorem for -Yang-Mills connections on .
Paper Structure (8 sections, 14 theorems, 96 equations)

This paper contains 8 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Connections and curvatures on vector bundles
  4. Curvatures and killing vector fields on $\mathbb{C}P^n$
  5. The stability and a gap theorem of $F$-Yang-Mills functional on $\mathbb{C}P^n$
  6. The second variation of $F$-Yang-Mills functional
  7. The estimation of the second variations under specific variation fields
  8. A gap theorem

Key Result

Theorem 1.1

(i) If $(2+\frac{4}{n})F"(x)x+(n+1)F'(x)<0$, then the $F$-Yang-Mills connection is unstable on $\mathbb{C}P^n$. (ii) Assume $(2+\frac{4}{n})F"(x)x+(n+1)F'(x)=0$ and $\nabla$ is a weakly stable $F$-Yang-Mills connection. Then there exists $\sigma\in\Omega^0(\mathfrak{g}_E)$ such that where $g$ is the Fubini-Study metric on $\mathbb{C}P^n$ and $J$ is the almost complex structure.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • ...and 12 more