2017/04/05 by Mihajlo Cekić, Cekić, Mihajlo · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1704.01362
openalex publication_date 2017/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of identifying a unitary Yang-Mills connection\n\∇ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of\nthe connection Laplacian \∇^*\∇ over compact Riemannian manifolds\nwith boundary. We establish uniqueness of the connection up to a gauge\nequivalence in the case of trivial line bundles in the smooth category and for\nthe higher rank case in the analytic category, by using geometric analysis\nmethods and essentially only one measurement.\n Moreover, by using a Runge-type approximation argument along curves to\nrecover holonomy, we are able to uniquely determine both the bundle structure\nand the connection, but at the cost of having more measurements. Also, we prove\nthat the DN map is an elliptic pseudodifferential operator of order one on the\nrestriction of the vector bundle to the boundary, whose full symbol determines\nthe complete Taylor series of an arbitrary connection, metric and an associated\npotential at the boundary.\n