| Abstract: |
| 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. 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. It was shown nby Graham
that an energy giving this (as gradient with respect to variation of
hypersurface embedding) arises 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 joint work with Andrew Waldron arXiv:1506.02723, CMP 2017 (arXiv:1603.07367),
Communications Contemp. Math. 2018 (arXiv:1611.08345)
Rod Gover |
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