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Ancient solutions to free boundary mean curvature flow

Theodora Bourni, Giada Franz

TL;DR

This work studies ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary, focusing on how unstable free boundary minimal hypersurfaces seed rigid ancient flows. Using a Morse-theoretic framework and a detailed linearization around a free boundary minimal hypersurface $\Sigma$, the authors construct an $I$-parameter family of ancient flows when $\Sigma$ has Morse index $I$, and show that all ancient solutions with exponential backward convergence to $\Sigma$ arise from this family. They also provide a second, more geometric construction of mean-convex ancient solutions via a free boundary mean-convex foliation built with an implicit-function theorem and barriers, giving a refined geometric description. Overall, the paper achieves a rigorous classification and rigidity theory for ancient solutions near free boundary minimal hypersurfaces in the convex-boundary setting, illuminating the structure of singularity-formation scenarios in free boundary MCF.

Abstract

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.

Ancient solutions to free boundary mean curvature flow

TL;DR

This work studies ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary, focusing on how unstable free boundary minimal hypersurfaces seed rigid ancient flows. Using a Morse-theoretic framework and a detailed linearization around a free boundary minimal hypersurface , the authors construct an -parameter family of ancient flows when has Morse index , and show that all ancient solutions with exponential backward convergence to arise from this family. They also provide a second, more geometric construction of mean-convex ancient solutions via a free boundary mean-convex foliation built with an implicit-function theorem and barriers, giving a refined geometric description. Overall, the paper achieves a rigorous classification and rigidity theory for ancient solutions near free boundary minimal hypersurfaces in the convex-boundary setting, illuminating the structure of singularity-formation scenarios in free boundary MCF.

Abstract

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.
Paper Structure (14 sections, 18 theorems, 191 equations, 1 figure)

This paper contains 14 sections, 18 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Further literature
  3. Future directions
  4. Plan of the paper
  5. Definitions and notation
  6. The Jacobi operator
  7. Weighted norms
  8. Graphs over a free boundary minimal hypersurface
  9. Graphs using a local parametrization
  10. Reversed Poincaré inequality for area-decreasing graphs
  11. Graphical mean curvature flow
  12. Constructing a family of solutions
  13. Rigidity of the constructed solutions
  14. Constructing mean-convex ancient solutions

Key Result

Theorem 1

Let $\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$. Moreover, any ancient solution with sufficiently fast decay

Figures (1)

  • Figure 1: Schematic representation of the notation.

Theorems & Definitions (50)

  • Theorem 1: cf. \ref{['construction', 'uniqueness']}, \ref{['COR']}
  • Theorem 2: cf. \ref{['MCconstruction', 'COR']}
  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of \ref{['prop:TimeSliceEstimates']}
  • Proposition 3.5: cf. AngenentDaskalopoulosSesum2019, BrendleChoi2019
  • ...and 40 more