Riesz continuity of the Atiyah-Singer Dirac operator under perturbations\n of local boundary conditions

2017/03/19 by Lashi Bandara, Andreas Rosén, Bandara, Lashi +1
Mathematics · #35J46 #35J56 #42B37 #58J05 #58J30 #58J32 #58J37 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1703.06548

openalex publication_date 2017/03/19 · openalex created_date 2023/03/15 · openalex updated_date 2026/07/28

Abstract

On a smooth complete Riemannian spin manifold with smooth compact boundary,\nwe demonstrate that the Atiyah-Singer Dirac operator \D _{mathcal B}\nin \L² depends Riesz continuously on \L^\∞\nperturbations of local boundary conditions mathcal B. The Lipschitz bound\nfor the map mathcal B \→ \D _{mathcal B}(1 +\n\D _{mathcal B}²)^-\(1)/(2) depends on Lipschitz smoothness\nand ellipticity of mathcal B and bounds on Ricci curvature and its first\nderivatives as well as a lower bound on injectivity radius. More generally, we\nprove perturbation estimates for functional calculi of elliptic operators on\nmanifolds with local boundary conditions.\n

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