Abstract: Inspired by Colding–Minicozzi's uniqueness theorem for asymptotic cones of Ricci-flat manifolds with Euclidean volume growth, obtained via monotone quantities, we construct analogous monotone quantities in the setting of linear volume growth. Using the average gradient estimate technique of Colding–Minicozzi and the level-set method of Munteanu–Wang, we show that these monotone quantities have applications to the topology of 3-manifolds with nonnegative scalar curvature under mild regularity assumptions. In particular, we prove that any contractible such manifold is diffeomorphic to $R^3$, and that any handlebody admitting such a metric must have genus at most one. In this talk, I will explain the motivation behind the construction of these monotone quantities and how they constrain the topology. This is joint work with Zetian Yan.

Brief biography: Xingyu Zhu is currently a visiting assistant professor at Michigan State University. He obtained his PhD degree in 2022 at Georgia Tech under the supervision of Prof. Igor Belegradek. His research interests are metric geometry and geometric analysis.

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