On the evolution of curves via a function of curvature. I. The classical case

AbstractThe problem of curve evolution as a function of its local geometry arises naturally in many physical applications. A special case of this problem is the curve shortening problem which has been extensively studied. Here, we consider the general problem and prove an existence theorem for the classical solution. The main theorem rests on lemmas that bound the evolution of length, curvature, and how far the curve can travel