Steepest descent curves of convex functions on surfaces of constant curvature

Abstract

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any steepest descent curve of a quasi-convex function in D is less than or equal to the perimeter of D. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S^2, it is also proved the existence of G-curves, whose length is equal to the perimeter of their convex hull, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.