The total absolute curvature of open curves in $E^{3}$

Abstract

The total absolute curvature of open curves in $E^3$ is discussed. We study the curves which attain the infimum of the total absolute curvature in the set of curves with fixed endpoints, end-directions, and length. We show that if the total absolute curvature of a sequence of curves in this set tends to the infimum, the limit curve must lie in a plane. Moreover, it is shown that the limit curve is either a subarc of a closed plane convex curve or a piecewise linear curve with at most three edges. The uniqueness of the curves minimizing the total absolute curvature is also discussed. This extends the results in [Yokohama Math. J. 48 (2000), 83–96], which deals with a similar problem for curves in $E^2$.