Riemann Surfaces with Shortest Geodesic of Maximal Length.
Given a closed Riemann surface S together a group of its conformal automorphisms H ≅ Z22, it is known that there are Schottky uniformizations of S realizing H. In this note we proceed to give an explicit Schottky uniformizations for each of all different topological actions of Z22 as group of conformal automorphisms on a closed Riemann surface.
Let be a hyperbolic surface and let be a Laplacian eigenfunction having eigenvalue with . Let be the set of nodal lines of . For a fixed analytic curve of finite length, we study the number of intersections between and in terms of . When is compact and a geodesic circle, or when has finite volume and is a closed horocycle, we prove that is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between and is . This bound is sharp.
We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, Int. Press, 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order α, 0 < α < 1, and for uniformly convex functions.