# Conformal Geometry of Embedded Manifolds with Boundary from Universal Holographic Formulae

## TL;DR

This paper develops a universal holographic framework for conformal invariants and differential operators on embedded manifolds with boundary, generalizing classical curvature concepts and providing explicit formulae for anomalies and divergences.

## Contribution

It introduces a universal boundary calculus and constructs extrinsic Q-curvatures and conformal Laplacian powers using a holographic approach involving singular Yamabe problems.

## Key findings

- Constructed critical order local invariants and differential operators.
- Derived universal formulae for extrinsic Q-curvatures and boundary transgression curvatures.
- Provided explicit expressions for volume and hyper-area anomalies and divergences.

## Abstract

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulae for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.