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Ricci flows which terminate in cones

Brett Kotschwar

TL;DR

The paper analyzes complete Ricci flows on manifolds with ends asymptotic to cones, showing that if such a flow converges to a cone with quadratic curvature decay as t→0, the flow is forced to be a gradient shrinking soliton. The method hinges on a novel backward-uniqueness framework built from Carleman estimates for a PDE-ODE system on an evolving asymptotically conical end, together with a careful extension argument that propagates the soliton structure from the end to the entire manifold via real-analyticity. The results establish that terminally conical AC solitons are gradient, and that the soliton structure on the end extends globally, yielding a gradient shrinking soliton on M. This provides a sharp rigidity statement for terminally conical Ricci flows and strengthens the link between asymptotic cone behavior and global self-similarity in Ricci flow, with potential implications for singularity models and the classification of AC shrinkers. The work blends PDE techniques (Carleman estimates, backward uniqueness) with geometric analysis (solitons, analytic continuation) to derive a robust, globally-constrained picture of AC Ricci flows near singular times.

Abstract

We prove that a complete solution to the Ricci flow on $M\times [-T, 0)$ which has quadratic curvature decay on some end of $M$ and converges locally smoothly to the end of a cone on that neighborhood as $t\nearrow 0$ must be a gradient shrinking soliton.

Ricci flows which terminate in cones

TL;DR

The paper analyzes complete Ricci flows on manifolds with ends asymptotic to cones, showing that if such a flow converges to a cone with quadratic curvature decay as t→0, the flow is forced to be a gradient shrinking soliton. The method hinges on a novel backward-uniqueness framework built from Carleman estimates for a PDE-ODE system on an evolving asymptotically conical end, together with a careful extension argument that propagates the soliton structure from the end to the entire manifold via real-analyticity. The results establish that terminally conical AC solitons are gradient, and that the soliton structure on the end extends globally, yielding a gradient shrinking soliton on M. This provides a sharp rigidity statement for terminally conical Ricci flows and strengthens the link between asymptotic cone behavior and global self-similarity in Ricci flow, with potential implications for singularity models and the classification of AC shrinkers. The work blends PDE techniques (Carleman estimates, backward uniqueness) with geometric analysis (solitons, analytic continuation) to derive a robust, globally-constrained picture of AC Ricci flows near singular times.

Abstract

We prove that a complete solution to the Ricci flow on which has quadratic curvature decay on some end of and converges locally smoothly to the end of a cone on that neighborhood as must be a gradient shrinking soliton.
Paper Structure (37 sections, 31 theorems, 365 equations)

This paper contains 37 sections, 31 theorems, 365 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Outline of the proof
  4. Reframing the question as one of backward uniqueness
  5. Embedding the problem into one for a larger system
  6. A general backward uniqueness theorem
  7. A backward uniqueness theorem for terminally conical Ricci flows
  8. Self-similarity
  9. Extending the soliton structure from $V$ to $M$
  10. Asymptotically conical families of metrics
  11. Basic decay estimates on the metric
  12. Decay estimates on the conical distance function
  13. Carleman estimates on ends smoothly emanating from a cone
  14. Uhlenbeck's trick and the operator $D_{\tau}$
  15. A general divergence identity
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $g(t)$ be a smooth complete solution to the Ricci flow on $M\times [-1, 0)$. Assume that there is $\mathcal{R}> 0$, an end $V\subset M$, and a diffeomorphism $F:\mathcal{C}_{\mathcal{R}} \longrightarrow V$ such that $\tilde{g}(t) = F^*g(t)$ converges locally smoothly to $\hat{g}$ on $\mathcal{C} on $\mathcal{C}_{\mathcal{R}}\times [-1, 0)$ for some constant $K$. Then there is $f\in C^{\infty}(

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • Proposition 3.2
  • ...and 51 more