MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.2 Self-expanding planar tree networks (Mazzeo–Saez, 2007; Ilmanen–Neves–Schulze, 2014)23B41DF2B95E

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Mazzeo–Saez — 2007-04-24 (first arXiv submission).
  2. Ilmanen–Neves–Schulze — 2014-07-17 (first arXiv submission).
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(Regular expanding trees with prescribed rays) A regular planar network consists of smooth branches meeting only at triple junctions, with angles $120$ degrees at every junction. A self-expanding network $\Sigma \subset \mathbb R^2$ satisfies

\[ \vec k=\frac{X^\perp }{2} \]

on each open branch; its time evolution is $\Sigma _ t=\sqrt t\, \Sigma $, $t>0$, with the same angle condition. It is a network flow and is not a smooth immersed curve at its junctions. For finitely many distinct prescribed rays from the origin, connected tree-like expanders provide smoothings in this network sense. Existence for three rays, including uniqueness of the tree, and existence for more rays are recalled in [INS19], Section 7, printed p. 70, supplied PDF p. 32, with attribution there to Schnürer–Schulze and Mazzeo–Sáez respectively.

The branches can be described as geodesics for the conformal metric

\[ g=e^{|X|^2/2}(dX_1^2+dX_2^2), \]

with the junction balance condition. Noncompact ends approach their assigned rays with Gaussian decay. Two regular expanders with the same asymptotic rays coincide if they are in the same topological class: the comparison deformation must retain distinct junctions and the prescribed asymptotic ends, as in the source definition. This does not assert uniqueness across different network topologies. See [INS19], Lemmas 4.1 and 4.3 and Theorem 4.4, printed pp. 49–51, supplied PDF pp. 11–13.

The source uses $\vec k=\psi ^\perp $ and $\sqrt{2t}\, \psi $; our profile is $\Sigma =\sqrt2\, \psi $. Nontrivial networks arise from a geodesic boundary-value construction. The exact balanced special case

\[ \bigcup _{j=0}^2\{ r(\cos (2\pi j/3),\sin (2\pi j/3)):r\geq 0\} \]

is the stationary $Y$ network. It illustrates the junction condition.

The accompanying picture instead computes a nonstationary four-ended example, using the geodesic construction of [MS07], Sections 2–3. In Euclidean arclength $s$, write the tangent as $(\cos \theta ,\sin \theta )$. Then

\[ x'=\cos \theta ,\qquad y'=\sin \theta ,\qquad \theta '=\frac{-x\sin \theta +y\cos \theta }{2}. \]

Shoot the upper-right branch from $(x,y,\theta )=(a,0,\pi /3)$ toward the ray of angle $\pi /4$. Numerically, $a\approx 0.385405$ in our normalization. Reflections in the coordinate axes and the straight segment $[-a,a]\times \{ 0\} $ give the displayed tree, with triple junctions at $(\pm a,0)$ and asymptotic rays at angles $\pi /4+j\pi /2$, $j=0,1,2,3$. A quarter-turn gives the second tree. The two pictures are congruent by rotation but have different connections when the four asymptotic ends are held fixed. Both have four ends and no cycles. These are numerical ODE approximations, not closed formulas or a classification of all network expanders. The time control shows $\sqrt t\, \Sigma $, $t>0$; the $t=0$ view shows the nonregular initial cone only.

Two planar expanding trees with the same four diagonal asymptotic rays: a blue horizontal connection and an orange vertical connection, each with two 120-degree junctions.
Numerical ODE solution. Two numerical four-ended expanders approach the same diagonal rays. Each has two 120-degree triple junctions and a straight central edge. The time control applies Σₜ = √t Σ₁; t = 0 shows only the initial cone. The trees differ with the ends held fixed, but are related by a quarter-turn. Ends are truncated for display. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Networks are not smooth immersed curves at their junctions; genus is not a surface invariant here.

Three prescribed rays · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Planar tree with triple junctions
Embeddedness
Embedded: Embedded as a network: branches meet only at their prescribed triple junctions. This is not a smooth embedded curve at a junction.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
3
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.
Four or more prescribed rays · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Planar tree with triple junctions
Embeddedness
Embedded: Embedded as a network: branches meet only at their prescribed triple junctions. This is not a smooth embedded curve at a junction.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
Number of prescribed rays (at least 4)
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.

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References on this page

  1. [INS19] Tom Ilmanen, André Neves, and Felix Schulze. On short time existence for the planar network flow. Journal of Differential Geometry, 111(1):39–89, 2019. Citations use the published 51-page PDF; printed page = PDF page + 38. Source expander normalization is curvature vector = normal position, converted by dilation sqrt(2). URL: https://arxiv.org/abs/1407.4756, doi:10.4310/jdg/1547607687.All uses in this project ↩ 1 ↩ 2
  2. [MS07] Rafe Mazzeo and Mariel Saez. Self-similar expanding solutions for the planar network flow. 2007. Preprint version 1; Section 2, Proposition 2.3, and Section 3. The source profile has curvature vector equal to normal position; dilate by sqrt(2) for the catalogue normalization. URL: https://arxiv.org/abs/0704.3113, arXiv:0704.3113.All uses in this project ↩ 1
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