§ 5.2 Purely rotating curve-shortening solitons (Halldórsson, 2010; Halldórsson, 2011; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)3A1BA479B53D

Let $A$ be a nonzero skew-symmetric linear map on $\mathbb R^ n$, $n\geq 2$. For a unit vector $T$ write $P_ Tw=w-\langle w,T\rangle T$. Given $C(0)=p$ and $T(0)=q$ with $|q|=1$, solve

The solution is unique for these initial data and exists for every $s\in \mathbb R$: $|T|=1$ and $|C(s)|\leq |p|+|s|$ prevent finite-parameter blow-up. Thus it is a complete immersed curve of codimension $n-1$. Its associated eternal flow is

where $\ell $ is arclength at time $t$. The exponential is an exact evolution law for the ODE-defined profile, not a closed formula for that profile. We exclude the stationary straight lines in $\ker A$.

Both ends satisfy $|C(s)|\to \infty $ as $s\to \pm \infty $; consequently the immersion is proper and noncompact, with domain $\mathbb R$ and two ends. Its distance from the origin has a unique minimum. In the plane, $A=\omega J$ with $J(x,y)=(-y,x)$ gives the rotating spiral family. The initial data $C(0)=0$, $T(0)=(1,0)$ give the usual centrally symmetric Yin–Yang example. A logarithmic spiral is not its exact profile; the purely rotating case has different far-field behavior from the expanding rotators in 5.3.

For $\mathbb R^3$, let $A(x,y,z)=\omega (-y,x,0)$, $\omega >0$. If $T_ z(0)\ne 0$, the identity

shows that $z$ is strictly monotone. The curve is therefore embedded and is a graph over an interval of the rotation axis. This interval can be bounded or a half-line; a finite endpoint is approached exponentially in arclength. If $T_ z(0)=0$, the curve stays in a plane perpendicular to that axis. More generally, the projection to $\ker A$ lies in a line, and any nonconstant such projection is injective. No blanket embeddedness assertion is made for profiles with constant kernel projection or for invertible $A$ in even dimensions.

Source: [AAAW13], Section 4.1, equation (27), Lemmas 5.4 and 6.3, PDF pp. 11–13, 15–20, and Section 7.1, p. 23. Initial-value uniqueness is not a uniqueness theorem for prescribed ends. No general curvature bound, stability theorem, or Gaussian-area or entropy value is recorded here. The figures integrate the profile equation with $\omega =1$: one spatial example has $p=(1,0,0)$, $q=(0,0,1)$; a second shows the planar Yin–Yang profile. Both are truncated numerical ODE curves with exact rotation in the time control. Planar classification (Halldórsson). Halldórsson classifies complete planar curves moving by Euclidean similarities in [Halldorsson2012]. The nontrivial motions are translation, dilation, rotation, or simultaneous rotation and dilation. The translator is the grim reaper (Example 3.1); the homothetic expanders are Example 4.3. The shrinking cases are linked from Example 2.11. A translation term in a motion with rotation or dilation can be absorbed by changing its center; see Section 2, PDF pp. 2–5.

Here is the common construction for the rotating families. Identify $\mathbb R^2$ with $\mathbb C$, fix $A\ne 0$, $B\in \mathbb R$, and use $T=e^{i\theta }$, $N=iT$, $\vec H=kN$. Solve globally in arclength

Then $X'=T$, $k=u$, and $k=A\langle X,T\rangle +B\langle X,N\rangle $, equivalently $\vec H=(AiX+BX)^\perp $. The exact motion of this ODE-defined profile is

The time formula is explicit; the spatial profile generally is not.

For $B=0$ and fixed $A\ne 0$, Theorem 3.1 gives a one-parameter family, up to rotation, specified by the minimum distance $d\in [0,\infty )$ from the origin. One may take $X(0)=id$, $T(0)=1$, hence $(u(0),v(0))=(0,Ad)$. Every member is a complete properly embedded noncompact curve in $\mathbb R^2$, of codimension one and topology $\mathbb R$, with two ends. Each end winds infinitely often and escapes to infinity; its curvature tends to zero while its total absolute curvature is infinite. For $A>0$ the signed curvature changes sign once, so these curves are not globally convex. Reflection reverses $A$. The $d=0$ member is the centrally symmetric Yin–Yang soliton; other members need not have that symmetry. The evolution exists for all real time. The identity $|X|^2-2\theta /A=\text{constant}$ is exact, but is not a polar formula because $\theta $ is the tangent angle.

The product $X(\mathbb R)\times \mathbb R\subset \mathbb R^3$ is also an eternal rotating MCF solution, with the same angular speed. It is a complete properly embedded surface, of codimension one, genus zero, topology $\mathbb R^2$ and one end. Its principal curvatures are the planar curvature and zero. This product case is the $h=\infty $ case discussed in [Halldorsson2013], Section 4, PDF p. 7; it is not a finite-pitch helicoidal translator.

The classification and properness are in Theorem 3.1, Corollary 3.6 and Section 3, PDF pp. 5–7; Figures 8–9 supply visual references. No Gaussian-area value, entropy value, or flow-stability conclusion is recorded in these passages. The figure is a numerical finite arclength window of the ODE, with time and minimum-distance controls. The compact ancient curves in Example 5.10 are a different family.