THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY

doi:10.1007/s10473-025-0518-7

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2142-2170.doi: 10.1007/s10473-025-0518-7

THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY

Wanjun AI, Miaomiao ZHU^*

1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

收稿日期:2024-05-21 修回日期:2024-09-24 出版日期:2025-09-25 发布日期:2025-10-14

THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY

Wanjun AI, Miaomiao ZHU^*

1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2024-05-21 Revised:2024-09-24 Online:2025-09-25 Published:2025-10-14
Contact: ^*Miaomiao Zhu, E-mail: mizhu@sjtu.edu.cn
About author:Wanjun Ai, E-mail: wanjunai@swu.edu.cn
Supported by:
Ai's research was supported by the National Natural Science Foundation of China (12201515). Zhu's research was partially supported by the Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-02-E00087), the National Natural Science Foundation of China (12171314) and the Shanghai Frontier Science Center of Modern Analysis.

摘要/Abstract

摘要： In this paper, we study the Neumann boundary value problem of the Yang-Mills $\alpha$-flow over a 4-dimensional compact Riemannian manifold with boundary. We establish the short-time existence of the Yang-Mills $\alpha$-flow in the framework of functional analysis and derive long-time existence and convergence results of classical solutions to the Yang-Mills $\alpha$-flow, provided that the $\alpha$-energy of initial connection is below some threshold. We also prove the validity of the boundary version of small energy estimates, removal of isolated singularities, and energy lower bound result for non-flat Yang-Mills connections. These results lead to the bubbling convergence of a sequence of Yang-Mills $\alpha$-connections, and as an application, we demonstrate the existence of non-trivial Yang-Mills connections with Neumann boundary.

关键词: Yang-Mills flow, initial boundary value problem, blow-up analysis

Abstract: In this paper, we study the Neumann boundary value problem of the Yang-Mills $\alpha$-flow over a 4-dimensional compact Riemannian manifold with boundary. We establish the short-time existence of the Yang-Mills $\alpha$-flow in the framework of functional analysis and derive long-time existence and convergence results of classical solutions to the Yang-Mills $\alpha$-flow, provided that the $\alpha$-energy of initial connection is below some threshold. We also prove the validity of the boundary version of small energy estimates, removal of isolated singularities, and energy lower bound result for non-flat Yang-Mills connections. These results lead to the bubbling convergence of a sequence of Yang-Mills $\alpha$-connections, and as an application, we demonstrate the existence of non-trivial Yang-Mills connections with Neumann boundary.

Key words: Yang-Mills flow, initial boundary value problem, blow-up analysis

中图分类号:

58E15

Wanjun AI, Miaomiao ZHU. THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY[J]. 数学物理学报(英文版), 2025, 45(5): 2142-2170.

Wanjun AI, Miaomiao ZHU. THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2142-2170.

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THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY

THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY

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