Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation
Abstract
This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics H0,ε on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large Kähler limit. More precisely, the mean curvature of H0,ε decays exponentially in every Ck-norm. Moreover, if H1,ε denotes the exact Hermitian Yang--Mills metric and Hε=H0,ε−1H1,ε, then, after normalization, for every nonnegative integer k, there exist positive constants Ck and ck such that |Hε− Id|Ck≤ Ck e−ck/ε. The main analytic difficulty lies in the global C0-comparison. Obtaining C0-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.
Related
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BibTeX
@misc{fu2026limiting,
title = {{Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation}},
author = {Jixiang Fu and Dekai Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29554},
eprint = {2608.29554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29554},
}
RIS
TY - UNPB
TI - Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation
AU - Jixiang Fu
AU - Dekai Zhang
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29554
UR - https://arxiv.org/abs/2608.29554
AB - This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics H_0,ε on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large Kähler limit. More precisely, the mean curvature of H_0,ε decays exponentially in every C^k-norm. Moreover, if H_1,ε denotes the exact Hermitian Yang--Mills metric and H_ε=H_0,ε^−1H_1,ε, then, after normalization, for every nonnegative integer k, there exist positive constants C_k and c_k such that |H_ε− Id|_C^k≤ C_k e^−c_k/ε. The main analytic difficulty lies in the global C^0-comparison. Obtaining C^0-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.
ER -