Higher Willmore energies, Q-curvatures, and related global geometry problems.

Rod Gover
02/04/2019 , 15:30 | Sala 236.

The Willmore energy and its functional gradient (under variations of embedding) have recently been the subject of recent interest in both geometric analysis and physics, in part because of their link to conformal geometry. Considering a singular Yamabe problem on manifolds with boundary shows that these these surface invariants are the lowest dimensional examples in a family of conformal invariants for hypersurfaces in any dimension. The same construction and variational considerations shows that (on even dimensional hypersurfaces) the higher Willmore energy and its functional gradient are analogues of the integral of the celebrated Q-curvature conformal invariant and its function gradient (now with respect to metric variations) which is known as the Fefferman-Graham obstruction tensor (or the Bach tensor in dimension 4). In fact the link is deeper than this in that the Willmore energy we consider is an integral of an invariant that actually generalises the Branson Q-curvature. This is part of fascinating unifying picture that includes some interesting open problems in global geometry.