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Preservation of product structures under the Ricci flow with instantaneous curvature bounds

Mary Cook

TL;DR

This work addresses whether a complete Ricci flow solution that starts as a product on $M=\hat M\times\check M$ remains a product under a curvature bound $|\mathrm{Rm}| \le \epsilon/t$. The authors recast the problem as a uniqueness question for a coupled PDE-ODE system that measures product deviation via time-dependent projections and curvature-derived tensors, and they prove a max-principle-based uniqueness result that forces the deviation to vanish under a small-curvature regime. Key contributions include the construction of time-dependent projections $\hat P(t)$, $\check P(t)$, the endomorphisms $\mathcal{P}$ and $\bar{\mathcal{P}}$, and a closed system for $\mathbf{X}$ and $\mathbf{Y}$, together with an adaptation of the Huang-Tam-Liu-Székelyhidi maximum principle to Ricci flow. Consequently, there exists an $\epsilon(n)>0$ such that any complete Ricci flow on $M=\hat M\times\check M$ with initial product metric and $|\mathrm{Rm}|\le \epsilon/t$ preserves the product structure for all $t\in[0,T]$, providing a rigidity-type result in this curvature regime and informing stability considerations for near-product geometries.

Abstract

In this note, we prove that there exists a constant $ε>0$, depending only on the dimension, such that if a complete solution to the Ricci flow splits as a product at time $t=0$ and has curvature bounded by $\fracε{t}$, then the solution splits for all time.

Preservation of product structures under the Ricci flow with instantaneous curvature bounds

TL;DR

This work addresses whether a complete Ricci flow solution that starts as a product on remains a product under a curvature bound . The authors recast the problem as a uniqueness question for a coupled PDE-ODE system that measures product deviation via time-dependent projections and curvature-derived tensors, and they prove a max-principle-based uniqueness result that forces the deviation to vanish under a small-curvature regime. Key contributions include the construction of time-dependent projections , , the endomorphisms and , and a closed system for and , together with an adaptation of the Huang-Tam-Liu-Székelyhidi maximum principle to Ricci flow. Consequently, there exists an such that any complete Ricci flow on with initial product metric and preserves the product structure for all , providing a rigidity-type result in this curvature regime and informing stability considerations for near-product geometries.

Abstract

In this note, we prove that there exists a constant , depending only on the dimension, such that if a complete solution to the Ricci flow splits as a product at time and has curvature bounded by , then the solution splits for all time.
Paper Structure (9 sections, 10 theorems, 103 equations)

This paper contains 9 sections, 10 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Tracking the product structure
  3. Extending the projections
  4. Evolution equations
  5. Constructing a PDE-ODE system
  6. A general uniqueness theorem for PDE-ODE systems
  7. Proof of Theorem \ref{['main']}
  8. Vanishing of time derivatives
  9. Preservation of product structures

Key Result

Theorem 1.1

Let $(\hat{M}, \hat{g}_0)$ and $(\check{M}, \check{g}_0)$ be two connected Riemannian manifolds and let $M = \hat{M} \times \check{M}$ and $g_0 = \hat{g}_0 \oplus \check{g}_0$. Then there exists a constant $\epsilon = \epsilon(n) > 0$, where $n = \mathrm{dim} (M)$, such that if $g(t)$ is a complete then $g(t)$ splits as a product for all $t \in [0,T]$, i.e., there exist $\hat{g}(t), \check{g}(t)$

Theorems & Definitions (20)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 10 more