Conformal hypersurface geometry via a boundary Loewner–Nirenberg–Yamabe problem

Abstract

We develop an approach to the conformal geometry of embedded hyper- surfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner-Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first main result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a hypersurface conformal invariant which gener- alises to higher dimensions the Willmore invariant of embedded surfaces. We call this the obstruction density and for even dimensional hypersurfaces this is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solu- tion to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers deter- mine an explicit holographic formula for the obstruction density. We also use these to construct conformally invariant hypersurface functionals. On each even dimensional hypersurface they provide a higher dimensional generalisation of the Willmore energy functional. In particular the functional gradient, with respect to variation of embed- ding, has linear leading term and thus provides another generalisation of the Willmore equation. It agrees with the obstruction density equation at leading derivative order.