2004/04/01 by A. Rod Gover, Gover, A. Rod · 1 citation
Mathematics · Physics and Astronomy · #35Q99 (secondary) #53A30 (primary) #53A55 #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/0404004
openalex publication_date 2004/04/01 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We look at several problems in even dimensional conformal geometry based\naround the de Rham complex. A leading and motivating problem is to find a\nconformally invariant replacement for the usual de Rham harmonics. An obviously\nrelated problem is to find, for each order of differential form bundle, a\n``gauge'' operator which completes the exterior derivative to a system which is\nboth elliptically coercive and conformally invariant. Treating these issues\ninvolves constructing a family of new operators which, on the one hand,\ngeneralise Branson's celebrated Q-curvature and, on the other hand, compose\nwith the exterior derivative and its formal adjoint to give operators on\ndifferential forms which generalise the critical conformal power of the\nLaplacian of Graham-Jenne-Mason-Sparling. We prove here that, like the critical\nconformal Laplacians, these conformally invariant operators are not strongly\ninvariant. The construction draws heavily on the ambient metric of\nFefferman-Graham and its relationship to the conformal tractor connection and\nexploring this relationship will be a central theme of the lectures.\n