Limit geometry of complete projective special real manifolds

TL;DR

This paper investigates the asymptotic limit geometries of complete projective special real manifolds, classifying possible limits and analyzing their symmetry properties, especially under regular boundary conditions.

Contribution

It provides a classification of limit geometries of these manifolds and establishes symmetry bounds, advancing understanding of their geometric structure and boundary behavior.

Findings

Identifies all possible limit geometries as complete projective special real manifolds.

Establishes a lower bound for the symmetry group dimension of limit geometries.

Shows that with regular boundary behavior, limits are isomorphic to a specific semi-direct product.

Abstract

We study the limit geometry of complete projective special real manifolds. By limit geometry we mean the limit of the evolution of the defining polynomial and the centro-affine fundamental form along certain curves that leave every compact subset of the initial complete projective special real manifold. We obtain a list of possible limit geometries, which are themselves complete projective special real manifolds, and find a lower bound for the dimension of their respective symmetry groups. We further show that if the initial manifold has regular boundary behaviour, every possible limit geometry is isomorphic to $\mathbb{R}_{>0}\ltimes\mathbb{R}^{n-1}$.