Extrinsic Paneitz operators and $Q$-curvatures for hypersurfaces

TL;DR

This paper derives explicit formulas for extrinsic Paneitz operators and $Q$-curvatures for hypersurfaces, revealing new conformal invariants and relating them to known functionals, thereby advancing the understanding of extrinsic conformal geometry.

Contribution

It provides explicit formulas for extrinsic Paneitz operators and $Q$-curvatures, identifying key conformal invariants and connecting them to existing geometric functionals.

Findings

Explicit formulas for ${f P}_4$ and ${f Q}_4$ in general dimensions.

Identification of the invariant $ extbf{C}$ as a combination of $J_1$ and $J_2$.

Relation of the integrals of $J_i$ to Guven and Graham-Reichert functionals.

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.