Upper Bounds on the Second Laplacian Eigenvalue on the Projective Space

This seminar talk from the "Spectral Geometry in the clouds" series features Hanna Kim from the University of North Carolina at Chapel Hill discussing upper bounds on the second Laplacian eigenvalue on the projective space. Explore the isoperimetric inequality of the second eigenvalue on real projective space across all dimensions. Learn how Nadirashvili and Penskoi demonstrated in two dimensions that the second eigenvalue reaches its maximum when a sequence of eigenvalues approaches that of a disjoint union of a sphere and a real projective space with a specific ratio. Discover the conjecture that this two-dimensional result might be generalizable to all dimensions within the conformal class of the round metric. Follow Kim's proof of an upper bound when the metric degenerates to that of two projective spaces, including the construction of trial functions for variational characterization using a generalized Veronese map to higher dimensional spheres. The presentation also covers topological degree methods and their additional applications, representing joint work with R. Laugesen.