Details
| Original language | English |
|---|---|
| Article number | 107700 |
| Number of pages | 59 |
| Journal | Advances in mathematics |
| Volume | 384 |
| Early online date | 8 Apr 2021 |
| Publication status | Published - 25 Jun 2021 |
Abstract
For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.
Keywords
- Conformal geometry, Embedded manifolds with boundary, Minimal hypersurface asymptotics, Q and T-transgression curvatures, Willmore energies with boundary, Yamabe problem
ASJC Scopus subject areas
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In: Advances in mathematics, Vol. 384, 107700, 25.06.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Conformal geometry of embedded manifolds with boundary from universal holographic formulæ
AU - Arias, Cesar
AU - Gover, A. Rod
AU - Waldron, Andrew
N1 - Funding Information: C.A. would like to thank the hospitality of QMAP at U.C. Davis and the Department of Mathematics at King's College London during earlier stages of this project. A.W. was also supported by a Simons Foundation Collaboration Grant for Mathematicians ID 317562, and thanks the University of Auckland for warm hospitality. A.W. and A.R.G. gratefully acknowledge support from the Royal Society of New Zealand via Marsden Grants 16-UOA-051 and 19-UOA-008.
PY - 2021/6/25
Y1 - 2021/6/25
N2 - For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.
AB - For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.
KW - Conformal geometry
KW - Embedded manifolds with boundary
KW - Minimal hypersurface asymptotics
KW - Q and T-transgression curvatures
KW - Willmore energies with boundary
KW - Yamabe problem
UR - http://www.scopus.com/inward/record.url?scp=85103963565&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1906.01731
DO - 10.48550/arXiv.1906.01731
M3 - Article
AN - SCOPUS:85103963565
VL - 384
JO - Advances in mathematics
JF - Advances in mathematics
SN - 0001-8708
M1 - 107700
ER -