The evolution by the curvature flow of the least diameter of a closed curve.

Abstract

The curvature flow of a curve was steadly studied in a series of papers by M. Gage, R. Hamilton [2], [3] and M. Grayson [4], that were published in the late 1980’s. These works concern mainly the long time behavior of regular closed plane curves which deform in the direction of the curvature vectors. In general the curves shrink to a point, but it becomes more and more "round" which means that the curvature tends to a constant. In this paper we will see that in addition the minimum of the diameters of a curve decreases when it is undergone by the curvature flow action.