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

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices