Vol. 311, No. 2, 2021

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Conformal boundary operators, $T$-curvatures, and conformal fractional Laplacians of odd order

A. Rod Gover and Lawrence J. Peterson

Vol. 311 (2021), No. 2, 277–328
Abstract

We construct continuously parametrised families of conformally invariant boundary operators on densities. These generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang–Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with conformal Laplacian power operators, the boundary operators yield conformally invariant fractional Laplacian pseudodifferential operators on the boundary of a conformal manifold with boundary. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson $Q$-curvature and include higher-order generalisations of the mean curvature and the $T$-curvature of Chang–Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We discuss applications of our results and techniques in the setting of Poincaré–Einstein manifolds and also use our constructions to shed light on some conjectures of Juhl.

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conformal boundary operators, conformal fractional Laplacians, $T$-curvature, differential intertwinors