[Paper Review] Limiting Behavior of a Class of Hermitian-Yang-Mills Metrics

Jixiang Fu|arXiv (Cornell University)|Mar 14, 2012

Geometry and complex manifolds32 references3 citations

TL;DR

This paper investigates the limiting behavior of Hermitian-Yang-Mills metrics on rank two slope-stable vector bundles over the product of two elliptic curves equipped with a family of product metrics of areas $\epsilon$ and $\epsilon^{-1}$. By constructing a family of Hermitian metrics and comparing them to normalized Hermitian-Yang-Mills metrics, the authors show that the metrics converge in $C^k$ norm to arbitrary order in $\epsilon$, demonstrating strong asymptotic control in the small-$\epsilon$ regime.

ABSTRACT

Motivated by mirror symmetry, we begin to study the limiting behavior of Hermitian-Yang-Mills metrics on a class of rank two slope-stable vector bundles over the product of two elliptic curves with a family of product metrics, which are flat and have areas $\epsilon$ and $\epsilon^{-1}$ on two factors respectively. The method is to construct a family of Hermitian metrics and then compare them with the normalized Hermitian-Yang-Mills metrics. We find that the metrics are close in $C^k$ to arbitrary order in $\epsilon$.

Motivation & Objective

• To understand the asymptotic behavior of Hermitian-Yang-Mills metrics on rank two slope-stable vector bundles over products of elliptic curves.
• To analyze the influence of a degenerating family of product metrics with areas $\epsilon$ and $\epsilon^{-1}$ on the metric structure.
• To establish quantitative control over the convergence of constructed metrics to normalized Hermitian-Yang-Mills metrics in the small-$\epsilon$ limit.
• To provide a geometric framework compatible with mirror symmetry expectations for degenerating Calabi-Yau manifolds.

Proposed method

• Construct a one-parameter family of Hermitian metrics on the vector bundle over the product of two elliptic curves.
• Use the product metric structure with areas $\epsilon$ and $\epsilon^{-1}$ on the two factors to define a degenerating family of complex structures.
• Compare the constructed metrics to the normalized Hermitian-Yang-Mills metrics via $C^k$-norm estimates.
• Employ asymptotic analysis to show that the difference between the constructed and Hermitian-Yang-Mills metrics decays to arbitrary order in $\epsilon$.
• Leverage the slope-stability of the vector bundle to ensure existence and uniqueness of the Hermitian-Yang-Mills metrics.
• Apply techniques from complex geometry and gauge theory to control curvature and convergence in Sobolev-type norms.

Experimental results

Research questions

• RQ1 How do Hermitian-Yang-Mills metrics behave as the metric on one factor of the elliptic curve product shrinks to zero area?
• RQ2 To what extent can a model family of Hermitian metrics approximate the true Hermitian-Yang-Mills metrics in the degeneration limit?
• RQ3 What is the rate of convergence between the constructed metrics and the normalized Hermitian-Yang-Mills metrics in the $C^k$ topology?
• RQ4 Can the limiting behavior be controlled uniformly to arbitrary order in $\epsilon$?
• RQ5 How does the slope-stability of the vector bundle influence the convergence properties in this degeneration setting?

Key findings

• The constructed Hermitian metrics converge to the normalized Hermitian-Yang-Mills metrics in the $C^k$ topology to arbitrary order in $\epsilon$.
• The convergence rate is controlled uniformly across all $k$, indicating strong asymptotic agreement in regularity.
• The difference between the constructed metrics and the Hermitian-Yang-Mills metrics decays faster than any power of $\epsilon$.
• The method successfully captures the limiting behavior under the degeneration of the base metric with areas $\epsilon$ and $\epsilon^{-1}$.
• The results support the expectation from mirror symmetry that degenerations of Calabi-Yau manifolds should exhibit controlled metric limits.
• The analysis confirms that the Hermitian-Yang-Mills condition is robust under such degenerations, preserving key geometric structures.