JournalsjemsVol. 13, No. 4pp. 911–957

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

  • Erwann Aubry

    Université de Nice, France
  • Colin Guillarmou

    Ecole Normale Superieure, Paris France
Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity cover
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Abstract

For odd dimensional Poincaré–Einstein manifolds (X__n+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 ifinity (∂_X_, [h_0]). In a second time we relate the set of C__n-2_k+1(Λ_k_(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 file://Untitled1generalization of Q curvature for forms.

Cite this article

Erwann Aubry, Colin Guillarmou, Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 911–957

DOI 10.4171/JEMS/271