MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.3 Expanding smoothings of planar corners (Halldórsson, 2010; Ding, 2015)D2DB1F7A1178

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. Halldórsson — 2010-07-09 (first arXiv submission).
  2. Ding — 2015-03-09 (first arXiv submission).
D2DB1F7A1178

The convention throughout these entries is $\vec H=X^\perp /2$ for the time-one profile. Thus $M_ t=\sqrt t\, M$, $t>0$, is the corresponding flow. The positive weight for variational statements is $e^{|X|^2/4}$; minimizing this weighted area locally will be called $E$-minimizing.

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(Even graphical corner expanders) For every $\kappa >0$ there is a smooth even function $\phi _\kappa :\mathbb R\to \mathbb R$ satisfying

\[ \frac{\phi _\kappa ''}{1+(\phi _\kappa ')^2} +\frac{x\phi _\kappa '-\phi _\kappa }{2}=0, \qquad \phi _\kappa '(0)=0, \qquad \lim _{|x|\to \infty }\frac{\phi _\kappa (x)}{|x|}=\kappa . \]

The profile $\Gamma _\kappa =\{ (x,\phi _\kappa (x)):x\in \mathbb R\} $ is a smooth complete properly embedded noncompact curve in $\mathbb R^2$, with topology $\mathbb R$, reflection symmetry, and two ends asymptotic to the rays of $y=\kappa |x|$. It is $E$-minimizing against compact replacements. Its curvature vector is nonzero and points along the upward graph normal: writing $q=\phi _\kappa -x\phi _\kappa '$ gives $q'=-x(1+(\phi _\kappa ')^2)q/2$ and $\phi _\kappa ''=(1+(\phi _\kappa ')^2)q/2$, with $q(0)>0$.

The source gives the quantitative estimate, for $x>0$,

\[ 0\leq \phi _\kappa (x)-\kappa x \leq \min \left\{ \frac{2e(\kappa +2)}{x}e^{-x^2/4},\, 3\kappa +2\right\} . \]

The flow $\sqrt t\, \Gamma _\kappa $, $t\in (0,\infty )$, smooths the corner at time zero. These are ODE-defined profiles, not elementary closed formulas. Source: [Ding20], Section 4, equations (4.9)–(4.15), printed pp. 373–374, supplied PDF pp. 15–16. The figure shows a numerical ODE profile with $\phi (0)=1$. Taking the product with $\mathbb R$ gives an example in $\mathbb R^3$ with one zero principal curvature.

For the planar classification, including the stationary-line endpoint, see [Halldorsson2012], Theorem 6.1, PDF pp. 13–15. Rotation together with expansion gives the different family in Example 5.3.

Expanding smoothings of planar corners. The even time-one profile has tip height 1 and is integrated over |x| <= 6. Its asymptotic cone slope is determined by the ODE, not prescribed independently.
Numerical ODE curve. The even time-one profile has tip height 1 and is integrated over |x| <= 6. Its asymptotic cone slope is determined by the ODE, not prescribed independently. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Expanding corner curve · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Line ℝ
Embeddedness
Embedded: The graphical smoothing is properly embedded, as is its product with a line.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
2
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.
Product with a line · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Plane ℝ²
Embeddedness
Embedded: The graphical smoothing is properly embedded, as is its product with a line.
Compactness
Noncompact
Surface genus
0
Topological ends
1
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. [Ding20] Qi Ding. Minimal cones and self-expanding solutions for mean curvature flows. Mathematische Annalen, 376:359–405, 2020. Citations use the supplied published 47-page PDF; printed page = PDF page + 358. URL: https://arxiv.org/abs/1503.02612, doi:10.1007/s00208-019-01941-1.All uses in this project ↩ 1
  2. [Halldorsson2012] Höskuldur P. Halldórsson. Self-similar solutions to the curve shortening flow. Transactions of the American Mathematical Society, 364(10):5285–5309, 2012. URL: https://arxiv.org/abs/1007.1617v2, doi:10.1090/S0002-9947-2012-05632-7.All uses in this project ↩ 1
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