We classify locally compact, geodesically complete, 2-dimensional Hadamard spaces whose Tits ideal boundaries have the minimal diameter . Furthermore, we classify the universal covering spaces of certain 2-dimensional nonpositively curved spaces, which is an extension of the result obtained in the polyhedral case by W. Ballmann, M. Brin, and S. Barré.
"Asymptotic rigidity of Hadamard 2-spaces." J. Math. Soc. Japan 52 (4) 699 - 723, October, 2000. https://doi.org/10.2969/jmsj/05240699