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Finite extinction time of a family of homogeneous Ricci flows

Roberto Araujo

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.

Finite extinction time of a family of homogeneous Ricci flows

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 , the space of all -invariant positive scalar curvature metrics is contractible.

Paper Structure

This paper contains 9 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on homogeneous Ricci flows
  3. The unimodular Ricci flow
  4. The unimodular Ricci flow along $\theta$-adapted metrics
  5. Algebraic bounds for the unimodular Ricci curvature
  6. The Dynamical Alekseevskii Conjecture on $\theta$-adapted standard metrics
  7. On the regularity of the eigenvalues of homogeneous Ricci flows
  8. Dynamical estimates and the main theorem
  9. The set of $U\ltimes V$-invariant positive scalar curvature metrics

Key Result

Theorem A

Let $G= U\ltimes V$ be a Lie group with Lie algebra $\mathfrak{g} = \mathfrak{u} \ltimes_{\theta} V$, where $V$ is an abelian ideal, $\mathfrak{u}$ is a compact Lie algebra acting on $V$, and the representation $\theta \colon \mathfrak{u} \to \mathfrak{gl}(V)$ is via semisimple operators. Let $H$ be Then this family is Ricci flow invariant. Moreover, if the universal cover of $M$ is not diffeomorp

Theorems & Definitions (56)

  • Theorem A
  • Theorem B
  • Definition 2.1: $G$-homogeneous manifold
  • Remark 2.2: Reduction to Lie algebra level
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Unimodular Ricci flow
  • Corollary 2.6
  • Definition 3.1: Stable representation
  • Remark 3.2
  • ...and 46 more