Inverse Spectral Problem for Surfaces of Revolution

TL;DR

This paper proves that for a specific class of revolution surfaces, the spectral data uniquely determines the surface's shape, leveraging integrable geodesic flows and quantum normal forms.

Contribution

It establishes that isospectral simple surfaces of revolution are necessarily isometric, using spectral invariants derived from quantum Birkhoff normal forms.

Findings

Spectral invariants at meridian geodesics are established.

The metric of the surface is uniquely determined by the spectral data.

Isospectral simple surfaces of revolution are shown to be isometric.

Abstract

This paper concerns the inverse spectral problem for analytic simple surfaces of revolution. By `simple' is meant that there is precisely one critical distance from the axis of revolution. Such surfaces have completely integrable geodesic flows with global action-angle variables and possess global quantum Birkhoff normal forms (Colin de Verdiere). We prove that isospectral surfaces within this class are isometric. The first main step is to show that the normal form at meridian geodesics is a spectral invariant. The second main step is to show that the metric is determined from this normal form.