With respect to whom are you critical?

2019/03/26 by Jin-ichi Itoh, Itoh, Jin-ichi, Costin Vîlcu +3
Mathematics · #53C45 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:53C45

paper · pdf · doi:10.48550/arxiv.1903.10908

19 pages, 3 figures

arxiv created 2019/03/26 · arxiv updated 2019/03/27

Abstract

For any compact Riemannian surface S and any point y in S, Q_y^-1 denotes the set of all points in S, for which y is a critical point. We proved \citeBIVZ together with Imre Bárány that cardQ_y^-1 ≥ 1, and that equality for all y∈ S characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface S and any point y ∈ S, the following two main results. There exist an open and dense set of Riemannian metrics g on S for which y is critical with respect to an odd number of points in S, and this is sharp. CardQ_y^-1 ≤ 5 for the torus and cardQ_y^-1 ≤ 8g-5 if the genus g of S is at least 2. Properties involving points at globally maximal distance on S are eventually presented.

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