L² harmonic forms on complete special holonomy manifolds

2018/01/13 by Teng Huang, Huang, Teng
Mathematics · #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1801.04443

openalex publication_date 2018/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we consider L² harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form ω. The main result is that if (X,ω) is a complete G₂- ( or Spin(7)-) manifold with a d(linear) G₂- (or Spin(7)-) structure form ω, the L² harmonic 2-forms on X will be vanish. As an application, we prove that the instanton equation with square integrable curvature on (X,ω) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over (X,ω).

Citations

• Differential equations on riemannian manifolds and their geometric applications
• Gauge Theory in higher dimensions, II

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