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 ${\bf P}_4$ of the Paneitz operator and the corresponding extrinsic fourth-order $Q$-curvature ${\bf Q}_4$ in general dimensions. This result involves a series of obvious local conformal invariants of the embedding $M^4 \hookrightarrow X^5$ (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant $\mathcal{C}$. In turn, we identify $\mathcal{C}$ as a linear combination of two local conformal invariants $J_1$ and $J_2$. Moreover, a linear combination of $J_1$ and $J_2$ can be expressed in terms of obvious local conformal invariants of the embedding $M \hookrightarrow X$. This finally reduces the non-trivial part of the structure of ${\bf Q}_4$ to the non-trivial invariant $J_1$. For closed $M^4 \hookrightarrow {\mathbb R}^5$, 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 $M^4 \hookrightarrow X^5$ 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 $M^4 \hookrightarrow X^5$. 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.