On the Extrinsic Geometry of Conformally Embedded Hypersurfaces

The relationship between the boundary of a manifold and its interior is important for studying many problems in science, as it allows us to predict the behavior of certain problems that can be modeled by partial differential equations. We study bulk-boundary relationships for conformal manifolds. A key tool for analyzing conformal manifolds is tractor calculus. By comparing the conformal structure in the interior with that of the boundary, we provide a complete hypersurface tractor calculus and develop a conformally-invariant characterization of the extrinsic curvature of the embedded hypersurface. These tools provide a characterization of families of conformal manifolds with boundaries that are of particular interest to physicists: so-called Poincare--Einstein manifolds and Willmore manifolds. Furthermore, we produce a series of conformally-invariant hypersurface operators and curvatures in boundary dimension four and discuss generalizations of these objects to arbitrary dimension