# Monge-Amp\`ere exhaustions of almost homogeneous manifolds

## TL;DR

This paper studies plurisubharmonic exhaustions satisfying Monge-Ampère equations on almost homogeneous manifolds, extending classical results and establishing rigidity in complex spaces modeled on these examples.

## Contribution

It introduces a new family of examples of Monge-Ampère exhaustions on almost homogeneous manifolds, extending classical theories to mixed type cases.

## Key findings

- Exhaustions satisfy complex homogeneous Monge-Ampère equations.
- Extends classical results to new mixed type examples.
- Provides rigidity results for complex spaces modeled on these manifolds.

## Abstract

We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Amp\`ere equations. This extends to a new family of mixed type examples various classical results on parabolic spaces and complexifications of symmetric spaces. Rigidity results on complex spaces modeled on such new examples are given.