  Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov,
and others, we consider the normal matrix model with an arbitrary potential
function, and explain how the problem of finding the support domain for the
asymptotic eigenvalue density of such matrices (when the size of the matrices
goes to infinity) is related to the problem of Hele-Shaw flows on curved
surfaces, considered by Entov and the first author in 1990-s. In the case when
the potential function is the sum of a rotationally invariant function and the
real part of a polynomial of the complex coordinate, we use this relation and
the conformal mapping method developed by Entov and the first author to find
the shape of the support domain explicitly (up to finitely many undetermined
parameters, which are to be found from a finite system of equations). In the
case when the rotationally invariant function is $\beta |z|^2$, this is done by
Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized
normal matrix model, which deals with random block matrices that give rise to
*-representations of the deformed preprojective algebra of the affine quiver of
type $\hat A_{m-1}$. We show that this model is equivalent to the usual normal
matrix model in the large $N$ limit. Thus the conformal mapping method can be
applied to find explicitly the support domain for the generalized normal matrix
model.
