The Dirichlet-to-Neumann Map for Poincar\'e-Einstein Fillings

TL;DR

This paper investigates the non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings, describing its range via a tensor related to volume variation and constructing conformally invariant hypersurface invariants.

Contribution

It introduces a new framework for understanding the Dirichlet-to-Neumann map in Poincaré-Einstein geometry, including explicit formulas and invariants, especially in even and odd dimensions.

Findings

Range of the map described by a rank two tensor.

Constructed conformally invariant hypersurface invariants.

Established uniqueness of these invariants under certain conditions.

Abstract

We study the non-linear Dirichlet-to-Neumann map for the Poincar\'e-Einstein filling problem. For even dimensional manifolds the range of this non-local map is described in terms of a rank two "Dirichlet-to Neumann tensor" along the boundary determined by the Poincar\'e-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincar\'e-Einstein metrics. We construct natural "Dirichlet-to-Neumann hypersurface invariants" that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order $T$-curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equaling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincar\'e-Einstein fillings for odd dimensional manifolds with conformally flat boundary.