Rigidity of proper almost-homogeneous domains in positive flag manifolds

TL;DR

This paper classifies proper domains in positive flag manifolds associated with Hermitian symmetric spaces of tube type, showing a unique orbit-closure boundary structure and addressing a rigidity question for such manifolds.

Contribution

It provides a classification of proper domains with boundary orbit closure properties in Hermitian symmetric spaces of tube type, and confirms a rigidity conjecture for flag manifolds with $ heta$-positive structures.

Findings

Unique proper domain classification in Shilov boundaries

Addresses rigidity for flag manifolds with $ heta$-positive structures

Provides a positive answer to Limbeek and Zimmer's rigidity question

Abstract

We show that, inside the Shilov boundary of any given Hermitian symmetric space of tube type, there is, up to isomorphism, only one proper domain such that every point on its boundary belongs to the closure of an orbit under its automorphism group. This gives a classification of all closed proper manifolds locally modelled on such Shilov boundaries, and provides a positive answer, in the case of flag manifolds admitting a $\Theta$-positive structure, to a rigidity question of Limbeek and Zimmer.