Geometric Analysis and Topology Seminar

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π_2 - systolic inequalities for 3-manifolds with positive scalar curvature

Speaker: Kai Xu, Duke University

Location: Warren Weaver Hall 512

Date: Friday, March 29, 2024, 11 a.m.

Synopsis:

We discuss a new systolic inequality for 3-manifolds with positive scalar curvature. It is known that if a closed 3-manifold has scalar curvature at least 1 and has nonzero second homotopy group, then its spherical 2-systole is bounded from above by 8π. Moreover, in the rigidity case the manifold is isometrically covered by a round cylinder. Recently the following gap theorem is proved by the speaker: if the manifold is topologically not a quotient of the cylinder, then the 2-systole is bounded by an improved universal constant that is approximately 5.44π. We will introduce this result along with some of the motivations and related ideas. The proof uses Huisken and Ilmanen's weak inverse mean curvature flow, which is another main component of the talk.