Rigidity of flag manifolds

TL;DR

This paper proves the rigidity of quasiconformal and Sobolev mappings on certain flag manifolds and upper triangular matrix groups, confirming the Regularity Conjecture for these structures when dimension is at least 4.

Contribution

It establishes a rigidity theorem for mappings on flag manifolds and upper triangular groups, settling the Regularity Conjecture in these contexts for dimensions n ≥ 4.

Findings

Quasiconformal homeomorphisms are rigid on flag manifolds.

Sobolev mappings with nondegenerate Pansu differential are also rigid.

Results extend to complex and quaternionic cases.

Abstract

Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally Sobolev mappings with nondegenerate Pansu differential, are rigid when $n \geq 4$; this settles the Regularity Conjecture for such groups. This result is deduced from a rigidity theorem for the manifold of complete flags in $R^n$. Similar results also hold in the complex and quaternion cases.