2022/10/08 by Andreas Juhl, Juhl, Andreas
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Primary 35J30 53B20 53B25 53C18 #Secondary 35J70 35Q76 53C25 58J50
paper · pdf · doi:10.48550/arxiv.2210.03982
openalex publication_date 2022/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 P4 of the Paneitz operator and the corresponding extrinsic fourth-order Q-curvature \bf Q4 in general dimensions. This result involves a series of obvious local conformal invariants of the embedding M4 \hookrightarrow 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. Moreover, a linear combination of J1 and J2 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 Q4 to the non-trivial invariant J1. For closed M4 \hookrightarrow \mathbb 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 \hookrightarrow 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 M4 \hookrightarrow X5. 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.