[Paper Review] Rigidity at the boundary for conformal structures and other Cartan geometries

Charles Frances|ArXiv.org|Jun 5, 2008

Geometric Analysis and Curvature Flows22 references3 citations

TL;DR

This paper establishes rigidity results for conformal embeddings of Riemannian and pseudo-Riemannian manifolds into larger Cartan geometries, showing that in dimension ≥3, conformal boundaries arising from such embeddings are uniquely determined up to diffeomorphism. It proves that complete flat Riemannian manifolds (≠ℝⁿ) and hyperbolic manifolds with full limit sets are conformally maximal, and that conformally homogeneous Riemannian manifolds of dimension ≥3 are conformally maximal, implying intrinsic uniqueness of the conformal boundary structure beyond dimension two.

ABSTRACT

In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension $\geq 3$, there are rigidity properties for the topological boundary of such a conformal embedding. We get results of the same kind about general Cartan geometries.

Motivation & Objective

• To determine when a pseudo-Riemannian manifold admits a strict conformal embedding into a larger manifold of the same signature.
• To investigate whether the topological boundary of such an embedding is uniquely determined or can vary wildly.
• To establish that for Riemannian manifolds of dimension ≥3, the conformal boundary is intrinsically determined, in contrast to the 2D case.
• To extend these results to general Cartan geometries, including conformal, projective, and affine structures.
• To prove conformal maximality for large classes of geometric structures, such as complete flat and hyperbolic manifolds.

Proposed method

• Uses the framework of Cartan geometries to generalize conformal structures and study conformal embeddings.
• Applies geometric analysis techniques, including length estimates on curves in product spaces.
• Employs a comparison of distances in submanifolds using curves with unit-speed projections.
• Uses the Riemann mapping theorem as a counterexample in 2D to contrast with higher-dimensional rigidity.
• Applies the concept of conformal maximality: a manifold is conformally maximal if every conformal embedding is onto.
• Leverages properties of limit sets in hyperbolic geometry to characterize conformal maximality of quotients Γ\Hⁿ.

Experimental results

Research questions

• RQ1 Under what conditions does a pseudo-Riemannian manifold admit a strict conformal embedding into a larger manifold of the same signature?
• RQ2 Can two different conformal embeddings of the same manifold yield topologically distinct boundaries?
• RQ3 Why does rigidity in the conformal boundary structure emerge in dimension ≥3 but not in dimension 2?
• RQ4 What geometric or dynamical conditions ensure that a manifold is conformally maximal?
• RQ5 How does conformal maximality relate to other geometric maximality notions, such as projective maximality?

Key findings

• Every complete flat Riemannian manifold of dimension n ≥ 3 that is not conformally equivalent to ℝⁿ is conformally maximal.
• For a complete hyperbolic manifold M = Γ\Hⁿ with n ≥ 3, conformal maximality is equivalent to projective maximality and to the limit set ΛΓ being equal to Sⁿ⁻¹.
• Complete hyperbolic manifolds of finite volume and dimension ≥3 are conformally maximal.
• Conformally homogeneous Riemannian manifolds of dimension ≥3 are conformally maximal.
• In dimension ≥3, the conformal boundary of a manifold is intrinsically determined: any two conformal embeddings yield boundaries that are bilipschitz equivalent.
• The paper establishes a general rigidity result for Cartan geometries: conformal boundaries are uniquely determined up to diffeomorphism in dimension ≥3.