Informal dynamical systems discussion: Liouville Metrics and Spectral Rigidity
I will present a joint work with Joscha Henheik, Vadim Kaloshin and Amir Vig on the deformationa spectral rigidty of Liouville metrics. The Liouville metrics refer to a class of Riemannian metrics on the 2-torus of the form ds2 = (f(x) + g(y))(dx2+dy2). Liouville metrics are special partly because they are conjectured to be the only class of metrics whose geodesic flow is integrable in the Liouville-Arnold sense. We consider the Laplace spectrum of such metrics, i.e., the set of eigenvalues λ of the PDE –Δu = λu where Δ is the Laplace-Beltrami operator associated with the metric. We prove that, for generic f and g, if a deformation of the form (f(x) + g(y) + εU(x,y))(dx2+dy2) preserves the Laplace spectrum, then U = 0. I will review the background of this problem and explain some key ideas of the proof.