Determination of the Genus of Surfaces from the Spectrum of Schr\"odinger Operators attached to height functions. (An inverse spectral problem for surfaces)

TL;DR

This paper demonstrates that the spectrum of Schrödinger operators linked to height functions on surfaces can determine the surface's genus and convexity, advancing inverse spectral problem techniques for geometric topology.

Contribution

It introduces a method to recover the topology and convexity of surfaces from spectral data, independent of the surface's metric, using inverse spectral analysis and wave invariants.

Findings

Genus of surfaces can be determined from spectral data.

Convexity of genus-zero surfaces can be detected.

Results are metric-independent and apply to generic height functions.

Abstract

Using results on inverse spectral problems, in particular the so-called new wave invariants attached to a classical equilibrium, we show that it is possible to determine the Morse index of height functions. For compact Riemannian surfaces $M\subset \mathbb{R}^3$ this imply that we can retrieve the topology (via the genus). Our results are independent from the choice of a metric on $M$ and can be obtained from the choice of a 'generic' height-function. For surfaces of genus zero, diffeomorphic to a 2-sphere, the method allows to detect the convexity, or the local convexity of the surface. Keywords : Micro-local analysis; Schr\"odinger operators; Inverse spectral problems.