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Geometry and Topology Seminar: Kai Xu, Duke, Isoperimetric inequalities on closed surfaces

November 9, 2022 | 11:00 am - November 10, 2022 | 12:00 pm EST

Let Σ be a closed surface (i.e. a 2-dimensional Riemannian manifold) satisfying the following condition: the first eigenvalue of the elliptic operator -Δ+βK is nonnegative, where K is the Gauss curvature and β is a positive constant . This condition was mainly motivated by the studies of positive scalar curvature in dimension three, and soon showed its own interest. This condition can be understood as a global and weak condition on the positivity of curvature. Therefore, it is natural to ask whether the surface Σ in question shows any geometric characteristic of positive curvature. We try to answer this problem in the perspective of isoperimetric inequalities. The main results are as follows. When β>1/2, the Cheeger constant of Σ is bounded from below in terms of the diameter. When β>1/4, the homogeneous isoperimetric ratio of Σ is bounded from below in terms of the diameter and total area of Σ. We also discuss the optimality of the bounds on β that appeared in the results.


November 9, 2022 | 11:00 am EST
November 10, 2022 | 12:00 pm EST
Event Category:


SAS 4201