Limiting Behavior of a Class of Hermitian-Yang-Mills Metrics, I

TL;DR

This paper investigates the asymptotic behavior of Hermitian Yang-Mills metrics on certain stable vector bundles over a product of elliptic curves as a parameter tends to zero, providing explicit constructions and estimates.

Contribution

It introduces explicit Hermitian metrics and compares them with HYM metrics, establishing higher order estimates and bounds in the limit of the parameter.

Findings

Constructed explicit Hermitian metrics on vector bundles.

Derived higher order estimates assuming $C^0$-estimate.

Provided bounds for the $C^0$-norm of HYM metrics.

Abstract

This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with K\"ahler metrics $\omega_\epsilon$ when $\epsilon\to 0$. Here $\omega_\epsilon$ are flat and have areas $\epsilon$ and $\epsilon^{-1}$ on the two elliptic curves respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the $C^0$-estimate is provided. We also get the estimate of the lower bound of the $C^0$-norm. If the desired estimate of the upper bound of the $C^0$-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in $\epsilon$ in any $C^k$ norms.