Chordal Loewner Equation

The aim of this survey paper is to present a complete direct proof of a well celebrated cornerstone result of Loewner Theory, originally due to Kufarev et al. [16]. It states that the family of the hydrodynamically normalized conformal self-maps of the upper-half plane onto the complement of a gradually erased slit satisfies, under a suitable parametrization, the chordal Loewner differential equation. The proof is based solely on basic theorems of Geometric Function Theory combined with some elementary topological facts, and it does not require any advanced technique