Apparaît dans la collection : Symmetry in Geometry and Analysis

An important special case of symmetry breaking differential operators are intertwining operators for spherical principal series representations which map smooth functions on a sphere to smooth functions on an equatorial subsphere. The symmetry is broken since equivariance holds true only for the subgroup of the conformal group of the big sphere which leaves the small sphere invariant. These operators interpolate between conformal powers of the Laplacian on both spheres. There are far-reaching analogs of these construction in the context of Riemannian manifolds $X$ with boundary $M$. We describe recent progress on such constructions in conformal differential geometry. The constructions rest on the solution of a singular version of the Yamabe problem. Among many other things, this leads to the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvatures (generalizing Branson’s $Q$-curvatures). The extrinsic conformal Laplacians are conformally covariant generalizations of so-called GJMS-operators (higher-order analogs of Yamabe and Paneitz operators). The extrinsic $Q$-curvatures are linked to conformal invariants of hypersurfaces. The presentation will rest on work with Bent Ørsted.