常曲率曲面上凸函数的最速下降曲线
Steepest descent curves of convex functions on surfaces of constant curvature
摘要：
设S是常曲率K =+1或-1的完备曲面，即球面S^2或Lobachevskij平面L^2，D是S的有界凸子集。如果S = S^2，则还假设直径（D）< pi/2。证明了D中拟凸函数的任何最速下降曲线的长度都小于或等于D的周长。这个上界实际上是为G-曲线类证明的，G-曲线是一个自然包含所有最速下降曲线的曲线族。当S = S^2时，证明了G-曲线的存在性，其长度等于其凸包的周长，表明上述估计确实是最优的.所得结果推广了Manselli和Pucci关于欧氏平面上最速下降曲线的定理.
英文摘要：
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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