Steepest descent curves of convex functions on surfaces of constant curvature

Let S be a complete surface of constant curvature K = ±1, i.e., S ² or л ², and Ω ? S a bounded convex subset. If S = S ², assume also diameter(Ω) < π/2. It is proved that the length of any steepest descent curve of a quasi-convex function in Ω is less than or equal to the perimeter of Ω. 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, the existence of G-curves, whose length is equal to the perimeter of their convex hull, is also proved, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.