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Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

J. M. Daniels-Holgate, Or Hershkovits

TL;DR

This work proves backward uniqueness for mean curvature flow past isolated, multiplicity-one, asymptotically conical singularities: if two smooth flows coincide at the first singular time, they coincide for all earlier times, without assuming global self-similarity or global AC behavior. The authors develop a novel Carleman-analytic framework that coherently handles the smooth region, the conical end, and the shrinking core near the singularity, combining a conditional backwards-uniqueness theorem with a three-stage rate-of-convergence analysis to show exponential decay of the height function between two flows. A key contribution is lower-bounding convergence across multiple geometric regimes, via global Carleman estimates tailored to roughly conical regions and a core shrinker analysis that leverages inverse-problem stability. As an immediate application, the paper derives backwards-uniqueness for low-entropy flows in $\mathbb{R}^4$, illustrating the practical impact on singularity analysis and the broader study of geometric flows with singularities.

Abstract

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces $M^1_t$ and $M^2_t$ encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time $T$, and if $M^1_T=M^2_T$ then $M^1_t=M^2_t$ for every $t\in [0,T]$. This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in $\mathbb{R}^4$ are backwards unique

Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

TL;DR

This work proves backward uniqueness for mean curvature flow past isolated, multiplicity-one, asymptotically conical singularities: if two smooth flows coincide at the first singular time, they coincide for all earlier times, without assuming global self-similarity or global AC behavior. The authors develop a novel Carleman-analytic framework that coherently handles the smooth region, the conical end, and the shrinking core near the singularity, combining a conditional backwards-uniqueness theorem with a three-stage rate-of-convergence analysis to show exponential decay of the height function between two flows. A key contribution is lower-bounding convergence across multiple geometric regimes, via global Carleman estimates tailored to roughly conical regions and a core shrinker analysis that leverages inverse-problem stability. As an immediate application, the paper derives backwards-uniqueness for low-entropy flows in , illustrating the practical impact on singularity analysis and the broader study of geometric flows with singularities.

Abstract

In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces and encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time , and if then for every . This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in are backwards unique

Paper Structure

This paper contains 14 sections, 21 theorems, 219 equations.

Table of Contents

  1. introduction
  2. Comparison to previous works
  3. Proof outline
  4. Organization
  5. Acknowledgements
  6. preliminaries
  7. Analytic structure of coinciding conical flows
  8. Validity of the AC singularity assumption
  9. A conditional backwards uniqueness theorem
  10. Lower bound on convergence for flows with asymptotically conical singularities
  11. Rate of convergence at smooth points
  12. A Carleman estimate for roughly conical flows in a ball
  13. Rate of convergence below the rough conical scale
  14. Rate of convergence at the core shrinker scale

Key Result

Theorem 4

Let $(M^i_t)_{t\in [0,t_0]}$ for $i=1,2$ be two mean curvature flows in $\mathbb{R}^{n+1}$, smooth in $\mathbb{R}^{n+1}\times [0,t_0] - \{(0,t_0)]\}$, having an AC singularity at $(0,t_0)$. If $M^1_{t_0}=M^2_{t_0}$ then $M^1_t=M^2_t$ for every $t\in [0,t_0)$.

Theorems & Definitions (48)

  • Conjecture 1: Backwards uniqueness conjecture past singularities for MCF in $\mathbb{R}^3$
  • Remark 1: On Backwards uniqueness and singularity analysis
  • Definition 2: Asymptotically conical singularity
  • Remark 3
  • Theorem 4: Backward uniqueness at isolated asymptotically conical singularities
  • Remark 5
  • Remark 6
  • Corollary 7: Backwards uniqueness of low entropy flows in $\mathbb{R}^4$
  • Remark 8: Contrasting uniqueness with backwards uniqueness
  • Remark 9: Best case scenario for a frequency analysis
  • ...and 38 more