Characterization of Lipschitz normally embedded complex curves and Lipschitz trivial values of polynomial mappings

Abstract

We study Lipschitz geometry of fibers of complex polynomial mappings from two points of view: the equivalence of inner and outer metrics of an algebraic curve and the existence of a locally bi-Lipschitz trivial fibre bundle structure over a subset of values of polynomial mappings. We prove that the affi ne part of a connected projective algebraic curve is Lipschitz normally embedded if and only if the following three conditions are satisfi ed: its affi ne part is connected, its affi ne part is locally Lipschitz normally embedded at each of its singular points; and its degree equals to the number of its points at infi nity. Moreover, we show that any Lipschitz trivial value of a real or complex polynomial mapping is a suspension of a regular value of properness of a polynomial mapping in fewer variables. Last, we show that this result cannot extend to rational mappings.