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Q-curvature, invariants, and higher Willmore energies for conformal hypersurfaces Gover, Rod
Description
The Willmore energy of a surface is a conformal measure of its failure to be conformally spherical. In physics the energy is variously called the bending energy, or rigid string action. In both geometric analysis and physics it has been the subject of great recent interest. We explain that its Euler-Lagrange equation is an extremely interesting equation in conformal geometry: the energy gradient is a fundamental curvature that is a scalar-valued hypersurface analogue of the Bach tensor (of dimension 4) of intrinsic conformal geometry. We next show that that these surface conformal invariants, i.e. the Willmore energy and its gradient (the Willmore invariant), are the lowest dimensional examples in a family of similar invariants in higher dimensions. A generalising analogue of the Willmore invariant arises directly in the asymptotics associated with a singular Yamabe problem on conformally compact manifolds. A result of Graham shows that an energy giving this (as gradient with respect to variation of hypersurface embedding) arises in a different way as a so-called "anomaly term" in a related renormalised volume expansion. We show that this anomaly term is, in turn, the integral of a local Q-curvature quantity for hypersurfaces that generalises Branson's Q-curvature by including coupling to the (extrinsic curvature) data of the embedding. This is associated to concormally invariant Laplacian power operators related to the celbrated GJMS operators, but which are coupled to the extrinsic curvature data of the embedding. This is joint work with Andrew Waldron arXiv:1506.02723, arXiv:1603.07367, arXiv:1611.08345
Item Metadata
Title |
Q-curvature, invariants, and higher Willmore energies for conformal hypersurfaces
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-05-22T11:10
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Description |
The Willmore energy of a surface is a conformal measure of its failure
to be conformally spherical. In physics the energy is variously called
the bending energy, or rigid string action. In both geometric analysis
and physics it has been the subject of great recent interest. We
explain that its Euler-Lagrange equation is an extremely interesting
equation in conformal geometry: the energy gradient is a fundamental
curvature that is a scalar-valued hypersurface analogue of the Bach
tensor (of dimension 4) of intrinsic conformal geometry.
We next show that that these surface conformal invariants, i.e. the
Willmore energy and its gradient (the Willmore invariant), are the
lowest dimensional examples in a family of similar invariants in
higher dimensions. A generalising analogue of the Willmore invariant
arises directly in the asymptotics associated with a singular Yamabe
problem on conformally compact manifolds. A result of Graham shows
that an energy giving this (as gradient with respect to variation of
hypersurface embedding) arises in a different way as a so-called
"anomaly term" in a related renormalised volume expansion. We show that this
anomaly term is, in turn, the integral of a local Q-curvature quantity
for hypersurfaces that generalises Branson's Q-curvature by including
coupling to the (extrinsic curvature) data of the embedding. This is
associated to concormally invariant Laplacian power operators related
to the celbrated GJMS operators, but which are coupled to the
extrinsic curvature data of the embedding.
This is joint work with Andrew Waldron arXiv:1506.02723, arXiv:1603.07367,
arXiv:1611.08345
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Extent |
37 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Auckland
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Series | |
Date Available |
2017-11-19
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0357990
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International