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Conformal hypersurface invariants and Bach-type Boundary Problems

Samuel Blitz, A. Rod Gover

TL;DR

The work advances conformal hypersurface geometry by constructing a new symmetric, trace-free boundary invariant that captures higher extrinsic data and encodes the Dirichlet-to-Neumann map for Poincaré–Einstein fillings. It develops a tractor-calculus–based framework to pair intrinsic hypersurface data with extrinsic fundamental forms, yielding conformally invariant boundary functionals and conserved currents within a variational setting. A key result is that Bach-flat manifolds with umbilic boundary admit a singular Yamabe metric that is formally Poincaré–Einstein, linking local boundary invariants to global asymptotics and providing a pathway to well-posed boundary problems via boundary terms. The paper also elucidates FG obstruction tensors, derives boundary pairings among higher fundamental forms, and discusses Extensions to FG-flat problems and potential higher-dimensional generalizations.

Abstract

Using variational considerations, we establish that there exists a new symmetric trace-free tensor conformal invariant of hypersurfaces embeddings in even dimensional conformal manifolds. This conformal invariant completes the family of conformal invariants known as conformal fundamental forms. The object has important links to global problems. In the context of the even dimensional boundary-value Poincaré--Einstein problem, the image of the Dirichlet--to--Neumann map is conformally invariant. Recent investigations established that this image is the pullback of a particular Riemannian invariant to the odd-dimensional boundary. We show here that, in fact, that image arises as the restriction of the new conformal invariant constructed here. As a consequence of the proof, we are able to construct several new global conformal invariants of the boundary. Finally, we use our variational results to establish that compact Bach-flat manifolds with umbilic boundary must admit a (formal to all orders) Poincaré--Einstein metric in the conformal class of its interior.

Conformal hypersurface invariants and Bach-type Boundary Problems

TL;DR

The work advances conformal hypersurface geometry by constructing a new symmetric, trace-free boundary invariant that captures higher extrinsic data and encodes the Dirichlet-to-Neumann map for Poincaré–Einstein fillings. It develops a tractor-calculus–based framework to pair intrinsic hypersurface data with extrinsic fundamental forms, yielding conformally invariant boundary functionals and conserved currents within a variational setting. A key result is that Bach-flat manifolds with umbilic boundary admit a singular Yamabe metric that is formally Poincaré–Einstein, linking local boundary invariants to global asymptotics and providing a pathway to well-posed boundary problems via boundary terms. The paper also elucidates FG obstruction tensors, derives boundary pairings among higher fundamental forms, and discusses Extensions to FG-flat problems and potential higher-dimensional generalizations.

Abstract

Using variational considerations, we establish that there exists a new symmetric trace-free tensor conformal invariant of hypersurfaces embeddings in even dimensional conformal manifolds. This conformal invariant completes the family of conformal invariants known as conformal fundamental forms. The object has important links to global problems. In the context of the even dimensional boundary-value Poincaré--Einstein problem, the image of the Dirichlet--to--Neumann map is conformally invariant. Recent investigations established that this image is the pullback of a particular Riemannian invariant to the odd-dimensional boundary. We show here that, in fact, that image arises as the restriction of the new conformal invariant constructed here. As a consequence of the proof, we are able to construct several new global conformal invariants of the boundary. Finally, we use our variational results to establish that compact Bach-flat manifolds with umbilic boundary must admit a (formal to all orders) Poincaré--Einstein metric in the conformal class of its interior.

Paper Structure

This paper contains 8 sections, 20 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Riemannian hypersurface geometry
  4. Conformal Geometry and Tractor Calculus
  5. Fefferman--Graham Obstruction Tensors and the Main Results
  6. Conserved Currents
  7. Conformally-Invariant Variational Problems on Manifolds with Boundary
  8. Fefferman--Graham Flat Obstructions to Poincaré--Einstein

Key Result

Theorem 1.1

For a hypersurface embedded in a conformal manifold $\Sigma \hookrightarrow (M^d,\boldsymbol{c})$ of even dimension $d$ there is natural conformal extrinsic invariant $\mathring{\underline{\overline{\rm{d}}}} \in \Gamma(\odot^2_\circ T^* \Sigma[3-d])$ such that $\mathring{\underline{\overline{\rm{d}

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Corollary 2.5
  • ...and 30 more