Extrinsic Paneitz operators and Q-curvatures for hypersurfaces

Abstract

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 P4 of the Paneitz operator and the corresponding extrinsic fourth-order Q-curvature Q4 in general dimensions. In the critical dimension n=4, this result yields a closed formula for the global conformal invariant ∫MQ4dvol (for closed M) and various decompositions of Q4, 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 M4↪X5 (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 J1 and J2. 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 J1 and J2 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 Q4 to the non-trivial invariant J1. For totally umbilic M, the invariants Ji vanish, and the formula for P4 substantially simplifies. For closed M4↪R5, we relate the integrals of Ji to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface M4↪X5 in general backgrounds. In this context, we find one further local conformal invariant J3. 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.