With respect to whom are you critical?

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 \cite{BIVZ} together with Imre B\'ar\'any that card$Q_y^{-1} \geq 1$, and that equality for all $y\in S$ characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface $S$ and any point $y \in S$, 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. Card$Q_y^{-1} \leq 5$ for the torus and card$Q_y^{-1} \leq 8g-5$ if the genus $g$ of $S$ is at least $2$. Properties involving points at globally maximal distance on $S$ are eventually presented.