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High Codimension Curve Shortening Flow with Free Boundary

Huy The Nguyen, Artemis A. Vogiatzi

Abstract

We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound $<2$. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.

High Codimension Curve Shortening Flow with Free Boundary

Abstract

We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound . Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.
Paper Structure (14 sections, 28 theorems, 161 equations)

This paper contains 14 sections, 28 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Curve shortening flow in high codimension
  3. Acknowledgements
  4. Evolution Equations
  5. Short Time Existence
  6. Dilation-Invariant Estimates
  7. Free-boundary adaptation of the dilation-invariant estimates
  8. Monotonicity and Entropy
  9. Type I Blowups and Self Shrinkers
  10. Type I Singularities
  11. Type II Blowups and Translators
  12. Planarity
  13. Hamilton argument
  14. Concluding argument

Key Result

Theorem 1.2

Let $\Omega\subset\mathbb R^{n+1}$ ($n\ge 2$) be a bounded, strictly convex domain with $C^2$ boundary. Let $\{\gamma_t\}_{t\in[0,T)}$ be a maximal free-boundary curve shortening flow in $\Omega$, starting from a properly embedded $C^2$-arc $\gamma_0$ that meets $\partial\Omega$ orthogonally. Assum Then exactly one of the following alternatives occurs:

Theorems & Definitions (49)

  • Definition 1.1: Free Boundary Curve Shortening Flow
  • Theorem 1.2: Long--time behaviour under low free--boundary entropy
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Interior evolution equations for curve shortening in $\mathbb{R}^{n+1}$
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Theorem 4.1: cf.Stahl1996, Theorem 3.1 and Section 6.3
  • Remark 4.2
  • ...and 39 more