A Calculus for Conformal Hypersurfaces and new higher Willmore energy functionals

TL;DR

This paper develops a calculus for conformal hypersurfaces, introduces new higher-dimensional Willmore energy functionals, and provides methods to compute invariants and differential operators related to conformal geometry.

Contribution

It introduces a new calculus for conformal hypersurfaces, constructs higher Willmore energy functionals, and develops tools for invariant computation and analysis.

Findings

Established a method to find distinguished ambient metrics for conformal hypersurfaces.

Developed a calculus of conformal hypersurface invariant differential operators.

Constructed new higher-dimensional Willmore energy functionals.

Abstract

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential operators and in particular, describe how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we develop new higher dimensional analogues of the Willmore energy for embedded surfaces. This complements recent progress on the existence and construction of such functionals.