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The aim of this short note is to investigate the geometry of weakly completesubdomains of Grauert type surfaces, i.e. open connected sets D, sitting insidea Grauert type surface X, which admit a smooth plurisubharmonic exhaustionfunction. We prove that they are either modifications of Stein spaces orGrauert type surfaces themselves and we apply these results to the special caseof Hopf surfaces.

Weakly complete domains in Grauert type surfaces