§ 2.11 Nonclosed shrinking curves dense in an annulus3A8C87C9FFF8

The nonclosed branch of the planar shrinking-curve classification is recorded in [Halldorsson2012], Theorem 5.1 and Section 5, PDF pp. 11–13. These are smooth complete immersed curves $X:\mathbb R\to \mathbb R^2$ of codimension one, in the normalization $\vec H=-X^\perp /2$. With arclength $s$, left normal $N=iT$, and signed curvature $k$, solve

Here $u(0)=r_{\min }/2$, $v(0)=0$, $\theta (0)=0$ and $0<r_{\min }<\sqrt2$. The phase orbit $(u,v)$ is periodic; write $P$ for its period and $\alpha =(2\pi )^{-1}\int _0^ P u(s)\, ds$. Choose $r_{\min }$ so that $\alpha $ is irrational. The period analysis gives $1/2<\alpha <1/\sqrt2$, and every irrational value in this interval occurs. This is an ODE construction with an irrational turning condition, not a closed formula.

The image is dense in the annulus $r_{\min }\leq |X|\leq r_{\max }$, where

Successive radial excursions are congruent by rotation through $2\pi \alpha $. The curvature is positive and periodic, with $k e^{-|X|^2/4}$ constant: local strict convexity does not imply global embeddedness. The curve is nonembedded and nonproper. Its parameter manifold is the noncompact line, with two intrinsic ends, despite its bounded image; neither end escapes to spatial infinity. Completeness follows from the global unit-speed parametrization. The annulus is the image closure, not a two-dimensional flow.

The exact evolution from this ODE-defined profile is $M_ t=\sqrt{-t}\, X(\mathbb R)$, $-\infty <t<0$, interpreted as a parametrized immersion. At every regular time its Gaussian area and entropy are infinite, with arclength counted on the parameter domain: an infinite-length curve lies in a bounded set, so every fixed Gaussian weight is bounded below there. This elementary consequence is separate from the source classification. No finite-entropy or stability theorem is claimed for this branch. For rational $\alpha =m/\ell $, the curve closes and is already recorded in Example 2.1; the circle and stationary line are Examples 2.9 and 2.10. The drawings numerically shoot for $\alpha =1/\sqrt2-1/100$ (a narrower annulus) and $\alpha =(\sqrt5-1)/2$ (a broader annulus). One complete excursion is highlighted, and its rotated continuations show how the image approaches density as the displayed parameter window grows. The same finite segment is shown at $t=-4,-1,-1/4,-1/16$, on a common spatial scale; the corresponding dilation factors are $2,1,1/2,1/4$. At time $t<0$, the annular radii are $\sqrt{-t}\, r_{\min }$ and $\sqrt{-t}\, r_{\max }$. The dashed circles are guides to the image closure, not additional components. The limit $t=0$ is not a regular immersed time slice. Finite precision cannot certify irrationality or depict the full dense image.