MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.7 Rotational graphical expanders over round conesA178E87D4BA0

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. Ding — 2015-03-09 (first arXiv submission).
A178E87D4BA0
D3CC5E948716

(Entire radial cone smoothings) For $n\geq 2$ and $\kappa >0$, consider

\[ M_\kappa =\{ (x,\phi _\kappa (|x|)):x\in \mathbb R^ n\} \subset \mathbb R^{n+1}, \]

where $\phi _\kappa $ is the smooth radial solution constructed with

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

The equation at $r=0$ is interpreted by smooth radial extension; it implies $\phi _\kappa ''(0)=\phi _\kappa (0)/(2n)$. This is an ODE construction, not a closed formula.

The hypersurface is smooth, complete, properly embedded, noncompact, rotationally symmetric, diffeomorphic to $\mathbb R^ n$, and $E$-minimizing against compact replacements. It has one conical end and nondecreasing height. The estimate

\[ 0\leq \phi _\kappa (r)-\kappa r \leq ((n+1)\kappa +2)\min \{ 1,r^{-1}\} ,\qquad r>0, \]

records the rate at which it approaches its cone. Its flow is $M_ t=\sqrt t\, M_\kappa $ for $t>0$, with initial cone $z=\kappa |x|$. Source: [Ding20], Section 4, equations (4.1), (4.7), and (4.8), printed pp. 371–373, supplied PDF pp. 13–15. The interactive figure shows a numerical rotational solution in $\mathbb R^3$ with $\phi (0)=1$.

Rotational graphical expanders over round cones. The R3 time-one profile has tip height 1 and is displayed up to radius 6. Its asymptotic cone is determined by this initial value.
Numerical ODE surface. The R3 time-one profile has tip height 1 and is displayed up to radius 6. Its asymptotic cone is determined by this initial value. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Radial graph surface · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Plane ℝ²
Embeddedness
Embedded: The smooth entire radial graph is properly embedded.
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.
Higher-dimensional radial graphs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Euclidean space ℝⁿ, n ≥ 3
Embeddedness
Embedded: The smooth entire radial graph is properly embedded.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
1
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.

Compare examples by these properties →

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
Cite this page

Comments

Comments are not enabled yet.