A survey of the GIT picture for the Yang-Mills equation over Riemann\n surfaces

2015/11/25 by Samuel Trautwein, Trautwein, Samuel
Mathematics · #14L24 #53C07 #53D20 #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1511.08122

openalex publication_date 2015/11/25 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to give a self-contained exposition of the\nAtiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an\nemphasis on the analogy to finite dimensional geometric invariant theory. The\nmain motivation is to provide a careful study of the semistable and unstable\norbits: This includes the analogue of the Ness uniqueness theorem for\nYang-Mills connections, the Kempf-Ness theorem, the Hilbert-Mumford criterion\nand a new proof of the moment-weight inequality following an approach outlined\nby Donaldson. A central ingredient in our discussion is the Yang-Mills flow for\nwhich we assume longtime existence and convergence.\n

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