On the embeddability of the homogeneous Ricci flow and its collapses

TL;DR

This paper investigates the embeddability of solutions to the homogeneous Ricci flow on flag manifolds, identifying which flows can be realized as embedded submanifolds and analyzing their collapses.

Contribution

It demonstrates that a large class of Ricci flow solutions can be realized as isometric embeddings, while some collapses cannot, providing explicit examples and a detailed flow analysis.

Findings

Existence of a global attractor with open interior for realizable metrics

Certain Ricci flow collapses cannot be embedded in fixed Euclidean spaces

Explicit examples of both realizable and non-realizable flow lines

Abstract

This article grew out of the urge to realize explicit examples of solutions for the Ricci flow as families of isometrically embedded submanifolds, together with its Gromov-Hausdorff collapses. To this aim, we consider the Ricci flow of invariant metrics in a class of flag manifolds. On the one hand, we contrast with a previous result in literature by presenting entire flow lines of invariant metrics realized as orbits of a fixed representation. Indeed, we prove that the subset of realizable metrics has a global attractor with open interior, containing an expressive family of complete flow lines. On the other hand, we prove that certain collapses cannot be realized in any fixed Euclidean space. We provide a detailed picture of the flow, including examples of both realizable and non-realizable flow lines and collapses.