Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ricci flow as functor

Alexander Plakhotnikov

TL;DR

The paper develops a functorial framework for Ricci flow by embedding it in an infinite-dimensional differentiable-stack setting, connecting the category of Riemannian manifolds (modulo diffeomorphisms) to singular Ricci flow spacetimes and then to geometric decompositions. It constructs a functorial chain $\mathsf{Riem}_{\mathrm{iso}} \to \mathsf{SingFlows} \to \mathsf{Dec}$ with a canonical initial data stack $\mathcal{X}=[\mathrm{Met}(M)/\mathrm{Diff}(M)]$, proving that the flow induces a stratification of this stack and preserves symmetries across singularities. The framework embeds surgery-like singularities into 4-dimensional cobordisms, yielding a robust, gauge-invariant description of evolution and decomposition, supported by examples including neck-pinch phenomena and Ricci solitons. This approach provides a principled, category-theoretic perspective on Ricci flow dynamics and their asymptotic geometric structure with potential implications for understanding moduli, symmetry, and topological changes under geometric flows.

Abstract

In this note we attempt to propose a categorical framework for the Ricci flow, treating it as a sequence of functors connecting the stack of Riemannian metrics to the category of geometric decompositions via singular flow spacetimes. To rigorize the domain of the flow, we adapt the definition of differentiable stacks to the site of Banach manifolds. We demonstrate that the Ricci flow defines a stratification of this stack.

Ricci flow as functor

TL;DR

The paper develops a functorial framework for Ricci flow by embedding it in an infinite-dimensional differentiable-stack setting, connecting the category of Riemannian manifolds (modulo diffeomorphisms) to singular Ricci flow spacetimes and then to geometric decompositions. It constructs a functorial chain with a canonical initial data stack , proving that the flow induces a stratification of this stack and preserves symmetries across singularities. The framework embeds surgery-like singularities into 4-dimensional cobordisms, yielding a robust, gauge-invariant description of evolution and decomposition, supported by examples including neck-pinch phenomena and Ricci solitons. This approach provides a principled, category-theoretic perspective on Ricci flow dynamics and their asymptotic geometric structure with potential implications for understanding moduli, symmetry, and topological changes under geometric flows.

Abstract

In this note we attempt to propose a categorical framework for the Ricci flow, treating it as a sequence of functors connecting the stack of Riemannian metrics to the category of geometric decompositions via singular flow spacetimes. To rigorize the domain of the flow, we adapt the definition of differentiable stacks to the site of Banach manifolds. We demonstrate that the Ricci flow defines a stratification of this stack.
Paper Structure (8 sections, 11 theorems, 19 equations)

This paper contains 8 sections, 11 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. Categorical framework
  5. The Category $\mathsf{SingFlows}$
  6. Stratification of the Infinite-Dimensional Stack
  7. Preservation of Symmetries
  8. Examples of Functorial Evolution

Key Result

Theorem 1.1

The assignment of a singular Ricci flow spacetime to a compact Riemannian 3-manifold constitutes a functor $\mathcal{F}: \mathsf{Riem}_{\mathrm{iso}} \longrightarrow\mathsf{SingFlows}$. Furthermore, the asymptotic limit of the flow defines a functor $\mathcal{A}: \mathsf{SingFlows} \longrightarrow\m

Theorems & Definitions (32)

  • Theorem 1.1: Theorem A: Functorial Construction
  • Theorem 1.2: Theorem B: Stratification of the Moduli Stack
  • Theorem 1.3: Theorem C: Conservation of Symmetries
  • Definition 2.1: Heb96 Definition 2.1
  • Definition 2.2: see, e.g. Ban16 Chapter I
  • Definition 2.3: Blo08 Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 22 more