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Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity

By Erwann Aubry and Colin Guillarmou

Abstract

Abstract. For odd dimensional Poincaré-Einstein manifolds (Xn+1, g), we study the set of harmonic k-forms (for k < n 2) which are Cm (with m ∈ N) on the conformal compactification ¯ X of X. This is infinite dimensional for small m but it becomes finite dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology Hk ( ¯ X, ∂ ¯ X) and the kernel of the Branson-Gover [3] differential operators (Lk, Gk) on the conformal infinity ( ∂ ¯ X,[h0]). In a second time we relate the set of Cn−2k+1 (Λk ( ¯ X)) forms in the kernel of d + δg to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of Q curvature for forms. 1

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.314.1940
Provided by: CiteSeerX
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