On the total absolute curvature of an immersed sphere (Q2519053)

In a recent paper, \textit{T. Ekholm} [Algebr. Geom. Topol. 6, 493--512 (2006; Zbl 1114.53049)] proved that for every \(\varepsilon > 0\) there is a sphere eversion through immersed spheres of the standard embedding of \(S^2\) into 3-space such that the total absolute curvature of the immersed spheres is always less than \(8\pi + \varepsilon\). It remains an open question whether this result is best possible. The present paper contains interesting results in relation to this question. As a particularly interesting thing it is proved that if the total absoute curvature does not exceed \(12\pi\) during an eversion, then the immersion must become non-simple at some point. (An immersion \(f\) in general position is called \textsl{simple} if for any irreducible self-intersection curve of \(f\) in 3-space, its two pre-image curves in the sphere are disjoint.)