Under asymptotic lower bound assumptions, all solutions of the Liouville equation on surfaces with nonnegative Ricci curvature are classified and the ambient manifold is shown to be rigid.
Cohn-Vossen,K¨ urzeste Wege und Totalkr¨ ummung auf Fl¨ achen, Compositio Math.2(1935), 69–133
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Classification results of Liouville equations and rigidity of Riemannian surfaces
Under asymptotic lower bound assumptions, all solutions of the Liouville equation on surfaces with nonnegative Ricci curvature are classified and the ambient manifold is shown to be rigid.