[Paper Review] Stability of solutions of certain extended Ricci flow systems

Michael Williams|arXiv (Cornell University)|Jan 16, 2013

Geometric Analysis and Curvature Flows4 references3 citations

TL;DR

This paper establishes dynamical stability for stationary solutions of four extended Ricci flow systems—Ricci flow coupled with harmonic map flow, Yang-Mills flow, and torsion heat flow—using maximal regularity theory and Simonett's results. It proves exponential convergence in Hölder norms for initial data near strictly linearly stable Einstein metrics with constant maps or warped product structures.

ABSTRACT

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped products), Ricci flow coupled with both harmonic map flow and Yang-Mills flow, and Ricci flow coupled with heat flow for the torsion of a metric-compatible connection. The methods used to prove stability follow a program outlined by Guenther, Isenberg, and Knopf, which uses maximal regularity theory for quasilinear parabolic systems and a result of Simonett.

Motivation & Objective

• To extend stability results for Ricci flow to extended systems involving additional geometric flows such as harmonic map, Yang-Mills, and torsion heat flows.
• To establish dynamical stability of stationary solutions in the context of warped product manifolds with Einstein fibers.
• To apply the Guenther-Isenberg-Knopf program to extended Ricci flow systems using maximal regularity and spectral theory.
• To prove exponential convergence of solutions in Hölder norms for initial data in a neighborhood of fixed points.
• To generalize known stability results for Einstein and Ricci-flat metrics to coupled geometric flows with additional structure.

Proposed method

• Uses curvature-normalized harmonic-Ricci-DeTurck flow to decouple the diffeomorphism invariance of Ricci flow.
• Applies maximal regularity theory for quasilinear parabolic systems to analyze long-time behavior.
• Employs Simonett's theory on center manifold reduction and spectral decomposition of the linearized operator.
• Imposes strict linear stability via the Lichnerowicz Laplacian with negative lower bound on the quadratic form.
• Analyzes the spectrum of the complexified linearization operator, separating stable and center subspaces.
• Constructs local $C^r$ center manifolds via implicit function theorems in Hölder spaces to describe asymptotic dynamics.

Experimental results

Research questions

• RQ1 Under what conditions is a constant map coupled with a strictly linearly stable Einstein metric dynamically stable under harmonic-Ricci flow?
• RQ2 How does Ricci flow behave on multiply-warped products with $μ_{\alpha}$-Einstein fibers under curvature-normalized DeTurck flow?
• RQ3 What is the long-term behavior of Ricci flow coupled with Yang-Mills and harmonic map flows on compact manifolds?
• RQ4 Can the Guenther-Isenberg-Knopf stability framework be extended to systems involving torsion evolution via heat flow?
• RQ5 What spectral and regularity conditions ensure exponential convergence of solutions in Hölder norms?

Key findings

• For Ricci flow coupled with harmonic map flow, solutions with initial data in a $(1+\theta)$-little-Hölder neighborhood of a constant map and strictly linearly stable Einstein metric converge exponentially fast in the $(2+\rho)$-Hölder norm.
• The convergence rate is exponential, with the solution approaching a limit where the map remains constant.
• In warped product settings, the system reduces to coupled equations for the base metric and warping functions, with stability preserved under $μ_{\alpha}$-Einstein assumptions.
• The linearized operator at the stationary solution has a spectral gap: $\sigma_{\mathrm{s}} \subseteq \{ z : \operatorname{Re} z \leq -\lambda_{\mathrm{s}} \}$, ensuring exponential decay of perturbations.
• A local $C^r$ center manifold exists near the fixed point, and solutions are exponentially attracted to it in the $\mathbb{X}_1$ norm.
• The estimate $\| \pi^{\mathrm{s}} \widetilde{\mathbf{g}}(\tau) - \gamma_d^r(\pi^{\mathrm{c}} \widetilde{\mathbf{g}}(\tau)) \|_{\mathbb{X}_1} \leq \frac{C_\alpha}{\tau^{1-\alpha}} e^{-\lambda\tau} \| \cdots \|_{\mathbb{X}_\alpha}$ quantifies the exponential attraction to the center manifold.