Extrinsic Paneitz operators and Q-curvatures for hypersurfaces.

For any hypersurface M of a Riemannian manifold X , recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q -curvatures. Here we derive explicit formulas for the extrinsic version P 4 of the Paneitz operator and the corresponding extrinsic fourth-order Q -curvature Q 4 in general dimensions. In the critical dimension n = 4 , this result yields a closed formula for the global conformal invariant ∫ M Q 4 d v o l (for closed M) and various decompositions of Q 4 , which are analogs of the Alexakis/Deser-Schwimmer type decompositions of global conformal invariants. These results involve a series of obvious local conformal invariants of the embedding M 4 ↪ X 5 (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant C. In turn, we identify C as a linear combination of two local conformal invariants J 1 and J 2. We also observe that these are special cases of local conformal invariants for hypersurfaces in backgrounds of general dimension. Moreover, in the critical dimension n = 4 , a linear combination of J 1 and J 2 can be expressed in terms of obvious local conformal invariants of the embedding M ↪ X. This finally reduces the non-trivial part of the structure of Q 4 to the non-trivial invariant J 1. For totally umbilic M , the invariants J i vanish, and the formula for P 4 substantially simplifies. For closed M 4 ↪ R 5 , we relate the integrals of J i to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface M 4 ↪ X 5 in general backgrounds. In this context, we find one further local conformal invariant J 3. Finally, we derive an explicit formula for the singular Yamabe energy of a closed M. The resulting explicit formulas show that it is proportional to the total extrinsic fourth-order Q -curvature. This observation confirms a special case of a general fact and serves as an additional cross-check of our main result. We carefully discuss the relations of our results to the recent literature, in particular to the work of Blitz, Gover and Waldron. [ABSTRACT FROM AUTHOR]