The Backward behavior of the Ricci and Cross Curvature Flows on SL(2,R)

TL;DR

This paper investigates the evolution of Ricci and cross curvature flows on SL(2,R) manifolds, showing that solutions typically start from sub-Riemannian geometries of Heisenberg type, resolving an open problem.

Contribution

It demonstrates the generic origin of maximal solutions from sub-Riemannian geometries, specifically of Heisenberg type, for flows on SL(2,R) manifolds, addressing an open question.

Findings

Maximal solutions originate at sub-Riemannian geometries of Heisenberg type

Provides a detailed analysis of flow behavior on SL(2,R)

Resolves an open problem from previous research

Abstract

This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL(2,R). We prove that, generically, a maximal solution originates at a sub-Riemannian geometry of Heisenberg type. This solves a problem left open in earlier work by two of the authors.