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The invariant theory for conformal hypersurfaces is studied by treating theseas the conformal infinity of a conformally compact manifold: For a givenconformal hypersurface embedding, a distinguished ambient metric is found(within its conformal class) by solving a singular version of the Yamabeproblem. Using existence results for asymptotic solutions to this problem, wedevelop the details of how to proliferate conformal hypersurface invariants. Inaddition we show how to compute the the solution's asymptotics. We also developa calculus of conformal hypersurface invariant differential operators and inparticular, describe how to compute extrinsically coupled analogues ofconformal Laplacian powers. Our methods also enable the study of integratedconformal hypersurface invariants and their functional variations. As a mainapplication we develop new higher dimensional analogues of the Willmore energyfor embedded surfaces. This complements recent progress on the existence andconstruction of such functionals.

A Calculus for Conformal Hypersurfaces and new higher Willmore energy functionals