Abstract
The aim of this short note is to investigate the geometry of weakly complete subdomains of Grauert-type surfaces, i.e., open connected sets D, sitting inside a Grauert-type surface X, which admit a smooth plurisubharmonic exhaustion function. We prove that they are either modifications of Stein spaces or Grauert-type surfaces themselves, and we apply these results to the special case of Hopf surfaces.
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