Limiting behavior of a class of Hermitian Yang-Mills metrics, I

摘要

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(a)hler metrics ωε when e → 0.Here,ωε are flat and have areas ε and e-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 C0-estimate is provided.We also get the estimate of the lower bound of the C0-norm.If the desired estimate of the upper bound of the C0-norm can be obtained,then it would be shown that these two families of metrics are close to arbitrary order in e in any Ck norms.

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