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arXiv · 2307.08470

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

Samuel Blitz·A. Rod Gover·Jarosław Kopiński·Andrew Waldron
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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.

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Samuel Blitz, A. Rod Gover, Jarosław Kopiński, Andrew Waldron. 2023-07-17. The Dirichlet-to-Neumann Map for Poincar\'e-Einstein Fillings. https://arxiv.org/abs/2307.08470

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