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Q-curvature, invariants, and higher Willmore energies for conformal hypersurfaces Gover, Rod


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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