Rigidity at the boundary for conformal structures and other Cartan geometries

TL;DR

This paper investigates the rigidity and maximality properties of conformal boundaries and Cartan geometries, revealing dimension-dependent rigidity phenomena and establishing foundational results for boundary embeddings.

Contribution

It introduces new rigidity results for conformal and Cartan geometries, especially regarding boundary behavior and maximality in higher dimensions.

Findings

Rigidity properties for the topological boundary in conformal embeddings in dimension ≥ 3

Results on conformal maximality and boundary embeddings

Extension of rigidity concepts to general Cartan geometries

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.