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Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

Giovanni Bellettini, Shokhrukh Kholmatov, Firdavsjon Almuratov

Abstract

We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.

Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

Abstract

We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.
Paper Structure (23 sections, 20 theorems, 215 equations, 22 figures)

This paper contains 23 sections, 20 theorems, 215 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Anisotropies
  5. Curves
  6. Tangential divergence of a vector field
  7. $\phi$-regular curves
  8. Networks and polygonal networks
  9. $\phi$-minimal networks
  10. $\phi$-curvature of an admissible network
  11. Computation of $\phi$-curvature
  12. Parallel networks
  13. $\phi$-curvature flow of admissible $\phi$-regular networks
  14. Parallel networks
  15. Proof of Theorem \ref{['teo:reconstruction']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

For any simple network $\mathscr{S}^0,$ there exists the unique $\phi$-curvature flow $\{\mathscr{S}(t)\}_{t\in [0,T^\dag)}$ starting from $\mathscr{S}^0$ on a maximal time interval $[0, T^\dag)$. Moreover, if $T^\dag< +\infty$, then some segment of $\mathscr{S}(t)$ vanishes as $t\nearrow T^\dag.$

Figures (22)

  • Figure 1: All possible self-shrinkers with one bounded phase.
  • Figure 2: The Wulff shape $B^\phi$ of sidelength $\frac{2}{\sqrt3}$, and its dual $B^{\phi^o}.$
  • Figure 3: A curve $\Gamma$ admitting a constant CH field $N$.
  • Figure 4: An admissible network containing $m$-junctions for $m=3,4,5,6.$
  • Figure 5: A $\phi$-regular admissible network (with $m$-junctions, $m=3,4$) consisting of the union of six polygonal curves $\Gamma_i$ with a CH field. At the triple junction $X$ we have $N_1 - N_2 + N_3=0,$ since $\Gamma_1$ and $\Gamma_3$ exit from $X$, while $\Gamma_2$ enters to $X.$ Thus, the balance condition \ref{['eq:balance_condition']} holds with $\sigma_1=\sigma_2=0$ and $\sigma_3=1$. Similarly, $N_2-N_3 + N_4+N_5 = 0$ at the quadruple junction $Y$ and $-N_4-N_5+N_6=0$ at the triple junction $Z.$
  • ...and 17 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Lipschitz $\phi$-regular curve
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: Curves with constant CH field
  • proof
  • Definition 2.5: Admissible network
  • Definition 2.6: $\phi$-regular networks and CH fields
  • ...and 52 more