Steepest descent curves of convex functions on surfaces of constant curvature

Details Steepest descent curves of convex functions on surfaces of constant curvature Steepest descent curves of convex functions on surfaces of constant curvature

2010-03-28

arxiv.org/abs/1003.5386v3

Mathematics

Classical Analysis and ODEs

23 pages, 3 figures; this is a revised version, with minor changes in definitions and adjustments in the proofs of Lemma 3.3 a

Scientific paper

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.

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