The object of the present
note is to indicate a derivation of the Löwner differential equations [1] based on the
derivation of an associated differential equation for Green’s function of the
variable region relative to the defining parameter. Decisive in our treatment is
the use of a certain normalized minimal positive harmonic function on the
variable region. In fact, our starting point was the feeling that the Poisson
kernel asserted its presence so strongly in the Löwner differential equations
that the concomitant presence of a normalized minimal positive harmonic
function on the variable region should appear naturally in the study of the
question. We shall see that this is the case. A technical advantage of the
present approach is that the “tip” lemmas of the classical proof are dispensed
with.
