Vadim Kaloshin, University of Maryland

Conférence Nirenberg du CRM en analyse géométrique: Can one hear the shape of a drum and deformational spectral rigidity of planar domains

Web site : http://www.crm.math.ca/Nirenberg2019/

M. Kac popularized the following question "Can one hear the shape of a drum?". Mathematically, consider a bounded planar domain $Omega subset mathbb R^2$ with a smooth boundary and the associated Dirichlet problem $Delta u+lambda u=0, u|_{partial Omega}=0$. The set of $lambda$'s for which this equation has a solution is called the Laplace spectrum of $Omega$. Does the Laplace spectrum determine $Omega$ up to isometry? In general, the answer is negative.

Consider the billiard problem inside $Omega$. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard inside $Omega$. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that a generic axially symmetric domain is dynamically spectrally rigid, i.e. cannot be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. The talk is a based on two separate joint works with J. De Simoi, Q. Wei and with J. De Simoi, A. Figalli.