Conformal Operators on Forms and Detour Complexes on Einstein Manifolds |
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Authors: | A Rod Gover Josef Šilhan |
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Institution: | (1) Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand;(2) Eduard Čech Center, Department of Algebra and Geometry, Masaryk University, Janáčkovo nám. 2a, 602 00 Brno, Czech Republic |
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Abstract: | For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential
complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the
cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that
generalise Branson’s Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here
we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds.
We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology
spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of
non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature
and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature
against the null space of the dimensional order conformal Laplacian of Graham et al. |
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