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

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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 Kähler metrics ωε when ε → 0. Here, ωε are flat and have areas ε and ε-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 ε in any Ck norms.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11871016, 11421061 and 11025103). The draft of the first six sections was finished in 2002 with the help of Professor Jun Li. The author thanks Jun Li for discussions on algebraic geometry and Professors Jiaxing Hong and Shing-Tung Yau on PDEs. The author also thanks Professors Guofang Wang, Qingxue Wang, Weiping Zhang and Xi Zhang for helpful discussions.

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  1. Institute of Mathematics, Fudan University, Shanghai, 200433, China

    Jixiang Fu

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  1. Jixiang Fu
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Correspondence to Jixiang Fu.

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Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

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Fu, J. Limiting behavior of a class of Hermitian Yang-Mills metrics, I. Sci. China Math. 62, 2155–2194 (2019). https://doi.org/10.1007/s11425-019-9512-6

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