Abstract: A constant mean curvature surface (CMC) is a critical point of the area functional subject to a volume constraint. Let M be a closed, three dimensional Riemannian manifold. The solution to the isoperimetric problem implies that, for each v between 0 and the volume of M, there is a constant mean curvature surface in M enclosing volume v. In this talk, we focus on the case where v is half the volume of M. We show that, when the metric on M is generic, there actually exist infinitely many distinct constant mean curvature surfaces cutting M into two pieces of equal volume. This solves a natural CMC version of Yau’s conjecture for generic metrics. This is joint work with Xin Zhou.

Bio: Liam Mazurowski is a postdoctoral researcher at Lehigh University. Before that, he was a postdoctoral researcher at Cornell University mentored by Xin Zhou. He received his Ph.D. from the University of Chicago in 2021 under the supervision of Andre Neves. His research specialties are differential geometry and geometric analysis.

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