Proper holomorphic self-maps of bounded domains with a weakly spherical boundary point

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Building on the results in Landucci (Adv Geom 12:515–523, 2012), we prove that if $$D$$ is a smoothly bounded weakly spherical domain of class $$\mathcal {C}^\infty $$ , with a weakly spherical point $$x_0 \in \partial D$$ with finite vanishing order $$\overline{q}$$ , then any proper self-map of $$D$$ which fixes $$x_0$$ is actually an automorphism. Moreover, we describe the geometry of weakly spherical points and we give a sufficient condition for the localization of proper maps between weakly spherical domains.