Generalized Willmore energies, Q-curvatures, extrinsic Paneitz
operators, and extrinsic Laplacian powers

J 2024

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON

Basic information

Original name

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Authors

BLITZ, Samuel Harris (840 United States of America, belonging to the institution), A Rod GOVER and Andrew WALDRON (guarantor)

Edition

Communications in Contemporary Mathematics, World Scientific Publishing Company, 2024, 0219-1997

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Singapore

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 1.600 in 2022

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1142/S0219199723500141

UT WoS

001157533200001

Keywords in English

Conformal geometry; extrinsic conformal geometry and hypersurface embeddings; conformally compact; Q-curvature; singular Yamabe problem; renormalized volume and anomaly; Willmore energy

Abstract

V originále

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.

Citovat

BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON. Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers. 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.

@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.

Detailed Information on Publication Record