A calculus for conformal hypersurfaces and new higher 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.

Renormalized volumes with boundary

“…This allows the boundary calculus for conformally compact manifolds of [GW14] to be applied to the singular Yamabe problem. Indeed, based on those methods, it was shown in [GW13, GW15,GW16b] that uniqueness properties of the solution to the singular Yamabe condition can be exploited to study the conformal geometry of embedded hypersurfaces. We shall show that in the singular Yamabe setting, the anomaly for embedded surfaces can be expressed in terms of the Euler characteristic χ Σ of the boundary and conformal hypersurface invariants integrated over Σ and ∂Σ.…”

“…The notion of a conformal hypersurface invariant is defined and studied in detail in [GW15,GW16b] (see also [Sta05,Vya13]) and refers to invariants of a hypersurface (or submanifold) Σ determined by the conformal embedding of Σ → M , in particular if P (Σ, g) is a hypersurface invariant determined by the embedding of Σ → (M, g) and P (Σ, Ω 2 g) := Ω w P (Σ, g), we shall denote the corresponding weight w conformal hypersurface invariant by P = [g ; P (Σ, g)] = [Ω 2 g ; Ω w P (Σ ; g)]. Important examples are the unit conormal to a hypersurfacen Σ := [g ; dσ/|dσ| g ] and the trace-free partII ab of the second fundamental form of Σ yieldingII Σ ab := [g ;II ab ]; these both have weight w = 1; we will drop the label Σ when the underlying hypersurface is clear from context.…”

“…We begin by solving the singular Yamabe condition (2.9). For that we employ the recursion of [GW15] (see the Appendix of [GW16b,GGHW15] for worked examples) and find that the densityσ = [g ;σ] with (7.10)σ =σ 1 −σ 4 3r 2 + 2…”

“…The integrand of the bulk´Σ term is the extrinsic Q-curvature of [GW15], and has been computed in [GW16b]. This yields for the bulk contribution 1…”

“…More recently it was realized that working with general conformally compact manifolds leads naturally to an extension to higher dimensional analogs of the Willmore equation [GW13, GW15,GW16b], and via volume expansions, energy functionals for these [Gra16,GW16a] (see also [GW16b,GGHW15] for an alternative approach). Here we show that there is tremendous gain in treating a new but related problem, namely solid infinite volume regions embedded in manifolds of the same dimension.…”