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@article{2385564,
author = {Blitz, Samuel Harris and Gover, A Rod and Waldron, Andrew},
article_number = {05},
doi = {http://dx.doi.org/10.1142/S0219199723500141},
keywords = {Conformal geometry; extrinsic conformal geometry and hypersurface embeddings; conformally compact; Q-curvature; singular Yamabe problem; renormalized volume and anomaly; Willmore energy},
language = {eng},
issn = {0219-1997},
journal = {Communications in Contemporary Mathematics},
title = {Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers},
url = {https://www.worldscientific.com/doi/10.1142/S0219199723500141},
volume = {26},
year = {2024}
}
TY - JOUR
ID - 2385564
AU - Blitz, Samuel Harris - Gover, A Rod - Waldron, Andrew
PY - 2024
TI - Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers
JF - Communications in Contemporary Mathematics
VL - 26
IS - 05
SP - 1-50
EP - 1-50
PB - World Scientific Publishing Company
SN - 02191997
KW - Conformal geometry
KW - extrinsic conformal geometry and hypersurface embeddings
KW - conformally compact
KW - Q-curvature
KW - singular Yamabe problem
KW - renormalized volume and anomaly
KW - Willmore energy
UR - https://www.worldscientific.com/doi/10.1142/S0219199723500141
N2 - Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled Q-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.
ER -
BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON. Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers. \textit{Communications in Contemporary Mathematics}. World Scientific Publishing Company, 2024, vol.~26, No~05, p.~1-50. ISSN~0219-1997. Available from: https://dx.doi.org/10.1142/S0219199723500141.
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