Closed curves in \(E^ 2\) and \(S^ 2\) whose total squared curvatures are almost minimal
Let \(\varepsilon\) be positive and let \(\gamma\) be a closed plane curve of length \(L\) with rotation number \(n\neq 0\) and curvature \(k\). The author proves that if \[ \int_\gamma k^2 ds<{(2\pi n)^2 \over L}+ {2\pi^2n^2 \varepsilon^2 \over L^3}, \] then \(\gamma\) lies between two concentric circles of radii \(r\) and \(R\) with \(|R-r|< \varepsilon\). A similar result is proved for closed curves in the 2-dimensional unit sphere.
scientific article; zbMATH DE number 1019777