The total absolute curvature of nonclosed curves in \(S^2\). II
A piecewise \(C^2\) curve in the unit sphere \(S^2\) is called an extremal curve if it has fixed endpoints, end-directions and length, and minimizes the total absolute curvature. In part I [Result. Math. 45, No. 1--2, 21--34 (2004; Zbl 1064.53002)], the authors proved that any extremal curve is a union of geodesic segments and small circular arcs with possible singularities at the endpoints. The present paper is a continuation of the above and the main results can be summarized in the following theorem: Any extremal curve of length less than \({{\pi }\over 2}\) is either a \(C^1\) curve consisting of one small circular arc and at most two geodesic segments, or a geodesic polygon consisting of at most three geodesic segments. If the extremal curve is a geodesic polygon with three sides, then two angles at the vertices must have different signs.
scientific article; zbMATH DE number 2119242