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Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

Sigurd B. Angenent, Dan Knopf

TL;DR

The paper develops a rigorous framework to study stability and instability of noncompact Ricci‑flat manifolds, emphasizing asymptotically conical ends and bounded geometry. By establishing that the Lichnerowicz Laplacian generates an analytic semigroup and has essential spectrum $(- abla o0]$, it identifies conditions yielding infinite‑dimensional unstable manifolds and, in Böhm metrics, uncountably many Ricci flows from unstable perturbations. The DeTurck trick is employed to recast Ricci flow as a parabolic system, whose linearization is governed by the Lichnerowicz Laplacian, enabling invariant manifold analysis and explicit construction of unstable directions. The analysis on Böhm doubly warped products shows infinitely many positive, simple, invariant eigenvalues, producing uncountably many geometrically distinct ancient Ricci flows. Overall, the work links spectral properties of geometric operators to nonlinear instability phenomena with explicit noncompact soliton examples.

Abstract

Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have uncountably many unstable perturbations.

Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

TL;DR

The paper develops a rigorous framework to study stability and instability of noncompact Ricci‑flat manifolds, emphasizing asymptotically conical ends and bounded geometry. By establishing that the Lichnerowicz Laplacian generates an analytic semigroup and has essential spectrum , it identifies conditions yielding infinite‑dimensional unstable manifolds and, in Böhm metrics, uncountably many Ricci flows from unstable perturbations. The DeTurck trick is employed to recast Ricci flow as a parabolic system, whose linearization is governed by the Lichnerowicz Laplacian, enabling invariant manifold analysis and explicit construction of unstable directions. The analysis on Böhm doubly warped products shows infinitely many positive, simple, invariant eigenvalues, producing uncountably many geometrically distinct ancient Ricci flows. Overall, the work links spectral properties of geometric operators to nonlinear instability phenomena with explicit noncompact soliton examples.

Abstract

Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have uncountably many unstable perturbations.

Paper Structure

This paper contains 51 sections, 15 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the paper
  4. An approach to DeTurck's trick
  5. Perturbed metrics and connections
  6. The Levi-Civita connections
  7. Lie derivative of the perturbed metric
  8. Perturbed curvature tensors
  9. Short time existence and smooth well-posedness
  10. Analytic semigroups
  11. Linearization near a fixed point
  12. Invariant manifolds, following Irwin
  13. Generation of an analytic semigroup
  14. $\mathscr L$ generates an analytic semigroup
  15. Eliminating the lower order terms
  16. ...and 36 more sections

Key Result

Proposition 1.3

Given integers $n_1, n_2 \geq 2$, there exists a Ricci flat metric on ${\mathbb R}^{n_1+1}\times \mathcal{S}^{n_2}$.

Theorems & Definitions (29)

  • Definition 1.1: Bounded geometry
  • Definition 1.2: Asymptotically flat
  • Proposition 1.3: Böhm B99
  • Lemma 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Lemma 4.1
  • proof
  • ...and 19 more