Finite extinction time of a family of homogeneous Ricci flows

TL;DR

This paper proves that certain noncompact homogeneous Ricci flows end in finite time, confirming a conjecture and showing the space of invariant positive scalar curvature metrics is contractible.

Contribution

It establishes finite extinction time for Ricci flows on a broad class of noncompact homogeneous manifolds and confirms the dynamical Alekseevskii conjecture for these spaces.

Findings

Ricci flow solutions have finite extinction time on these manifolds

The space of G-invariant positive scalar curvature metrics is contractible

Supports the dynamical Alekseevskii conjecture

Abstract

We show that for a broad family of noncompact homogeneous Riemannian manifolds, the corresponding homogeneous Ricci flow solutions have finite extinction time, thereby confirming the dynamical Alekseevskii conjecture for these spaces. As an application, we prove that on such homogeneous manifolds $G/H$, the space of all $G$-invariant positive scalar curvature metrics is contractible.