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The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings

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

Abstract

We study the non-linear Dirichlet-to-Neumann map for the Poincaré-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 Poincaré-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré-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 Poincaré-Einstein fillings for odd dimensional manifolds with conformally flat boundary.

The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings

Abstract

We study the non-linear Dirichlet-to-Neumann map for the Poincaré-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 Poincaré-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré-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 -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 Poincaré-Einstein fillings for odd dimensional manifolds with conformally flat boundary.
Paper Structure (8 sections, 12 theorems, 136 equations)

This paper contains 8 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Conformal Hypersurface Calculus
  4. Dirichlet-to-Neumann Tensors
  5. $T$-curvatures
  6. Even dimensional bulk
  7. Odd dimensional bulk
  8. Proof of Theorem \ref{['blunt']}

Key Result

Theorem 1.1

Let $(M_+^6,g^o)$ be a Poincaré--Einstein structure with conformal infinity $\Sigma$. Then the unique natural transverse order 5 section of $\mathring{\top}\!\!\odot^2\! \!T^*M[-3]|_\Sigma$ that is an invariant of the Poincaré--Einstein structure is given, up to a non-zero constant multiple, for a c Furthermore

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1
  • proof
  • Proposition 3.2: GPt Proposition 6.15
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 17 more