Classification results of Liouville equations and rigidity of Riemannian surfaces

Key Points

• This research aims to classify solutions to the Liouville equation on Riemannian surfaces with nonnegative Ricci curvature and to explore rigidity results for the manifold.
• Examined the Liouville equation under nonnegative Ricci curvature conditions.
• Utilized asymptotic lower bound assumptions for classification.
• Assessed the optimality of assumptions compared to classical finite total curvature criteria.
• Classified all solutions to the Liouville equation in the given context.
• Established rigidity results for the Riemannian surface under specific assumptions.
• Demonstrated that the assumptions used are optimal in certain scenarios.