Extrinsic Paneitz operators and Q-curvatures for hypersurfaces

TL;DR

This paper provides explicit formulas for extrinsic Paneitz operators and Q-curvatures on hypersurfaces, revealing their conformal invariance and decompositions, advancing understanding of extrinsic conformal geometry.

Contribution

It introduces explicit formulas for extrinsic Paneitz operators and Q-curvatures for hypersurfaces, including totally umbilic cases, and establishes their conformal invariance and decomposition properties.

Findings

Explicit formulas for extrinsic P_4 and Q_4.

Critical extrinsic P_4 and Q_4 are conformally invariant.

Decomposition of the critical extrinsic Q_4 analogous to known decompositions.

Abstract

For any hypersurface of a Riemannian manifold, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we announce explicit formulas for the extrinsic Paneitz operators P_4 and the corresponding extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension. Moreover, we state explicit formulas for the critical extrinsic P_4 and the total integral of the critical extrinsic Q_4 in the general case. This integral is a global conformal invariant. Finally, we establish an analog of the Alexakis-Deser-Schwimmer decomposition of the critical extrinsic Q_4.