Spectral rigidity and integrability for billiards and geodesic flows

In 1911, Hermann Weyl proved the remarkable asymptotic formula describing distribution of (large) eigenvalues of the Dirichlet Laplacian in a bounded domain Ω ⊂ RdN (λ) = (2π)−d ωd Vol(Ω) λd/2(1 + o(1))as λ → +∞. where N (λ) is the number of eigenvalues of the Laplacian spectrum, which are less than λ, ωd is a volume of the unit ball in Rd, Vol(Ω) is the volume of Ω, and the Laplace spectrum of a domain Ω is defined as the set of positive real numbers λ (with multiplicities) that satisfy the eigenvalue problem in Ω with Dirichlet boundary conditions.

This result motivated the title of a famous paper by M. Kac “Can you hear the shape of a drum?”. The question is: can the shape of a bounded domain O C Rd be determined by the Laplace spectrum? Two domains are called isospectral if they have the same eigenvalues. Consider the space of domains with a smooth boundary.

The existence of isospectral non-isometric domains is a well-known open question.The first goal of the project is to prove the local spectral rigidity for convex planar domains, i.e. for a smooth convex planar domain Ω the Laplace spectrum determines Ω locally. There are no nearby isospectral non-isometric domains with smooth boundary.

All of the these questions can also be posed for Riemannian manifolds.

The second goal is to prove the local rigidity for Riemannian manifolds with Anosov geodesic flows.The third goal is to prove local rigidity for integrable systems: geodesic flows on tori (resp. convex planar billiards). The goal is to prove that an integrable metric close to a Liouville metric is Liouville.

The second type is billiards inside smooth planar domains integrable near the boundary.

We shall prove that domains with integrable billiards belong to a finite-dimensional manifold.The focal goal of the project is to develop analytic tools to solve the local inverse problems for smooth planar convex domains and geodesic flows.