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On the dynamical behaviour of the generalized Ricci flow

Alberto Raffero, Luigi Vezzoni

TL;DR

This work studies the dynamical behavior of the generalized Ricci flow on compact manifolds with a fixed background closed 3-form $\hat{H}$ by employing the variational functional $\mu(g,b)=\lambda(g,\hat{H}+db)$. The authors establish a Lojasiewicz-Simon inequality for $\mu$ and implement a Nash–Moser gauge fixing to derive a DeTurck-type reformulation, enabling a gradient-flow analysis of the flow. The main results show dynamical stability: if $(\hat{g},0)$ is a local maximizer of $\mu$, solutions starting nearby exist globally and converge to a Ricci-flat metric modulo diffeomorphisms; conversely, non-maximizers yield dynamical instability in the form of nontrivial ancient solutions converging to $(\hat{g},0)$ as $t\to -\infty$. Overall, the paper extends stability results from Ricci flow to the generalized setting and connects the dynamics to a variational framework via $\mu$.

Abstract

Motivated by Müller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.

On the dynamical behaviour of the generalized Ricci flow

TL;DR

This work studies the dynamical behavior of the generalized Ricci flow on compact manifolds with a fixed background closed 3-form by employing the variational functional . The authors establish a Lojasiewicz-Simon inequality for and implement a Nash–Moser gauge fixing to derive a DeTurck-type reformulation, enabling a gradient-flow analysis of the flow. The main results show dynamical stability: if is a local maximizer of , solutions starting nearby exist globally and converge to a Ricci-flat metric modulo diffeomorphisms; conversely, non-maximizers yield dynamical instability in the form of nontrivial ancient solutions converging to as . Overall, the paper extends stability results from Ricci flow to the generalized setting and connects the dynamics to a variational framework via .

Abstract

Motivated by Müller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.

Paper Structure

This paper contains 3 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The functional $\mu$
  3. Proof of Theorem \ref{['main1']}

Key Result

Theorem 1.1

Let $(M,\hat{g})$ be a compact Ricci-flat manifold. Assume that $(\hat{g},0)\in C^\infty(M,S^2_{+}) \times C^\infty(M,\Lambda^2)$ is a local maximizer of $\mu(g,b) = \lambda(g,db)$. Then, there exists an open neighbourhood $\mathcal{U}$ of $(\hat{g},0)$ in $C^{\infty}(M,S^2_{+})\times dC^{\infty}(M,

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 3 more