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For $n\ge 2$, a round annulus $A_R=B_{2R}\setminus B_R$ is connected and obeys ${\int }_{A_R}|v-v_{A_R}{|}^2\le C(n)R^2{\int }_{A_R}|\nabla v{|}^2$. Scaling reduces the claim to a fixed annulus. In dimension one the annulus has two components, so the inequality fails with a single average; one needs a separate average on each component.

For a common semi-major axis $a$ and proper eccentricity $e_p$, randomly distributed proper longitudes of periapsis give an annulus of radii $a(1-e_p)$ and $a(1+e_p)$ about the point displaced from the star by minus $a$ times the forced eccentricity vector, to first order in orbital eccentricity.

In orientation-preserving annular coordinates $(\theta ,r)$ with orientation $d\theta \wedge dr$, the twist is $(\theta ,r)\mapsto (\theta +2\pi f(r),r)$, where $f$ increases from zero to one. The map is the identity outside this annulus and near its boundary. The orientation of the ambient topological surface, rather than the orientation assigned to the curve, fixes the handedness.

A vertical surface is a union of Seifert fibers over an arc or a closed curve in the base orbifold. Typical connected examples are an annulus and a torus.