Steklov-isospectral strictly convex planar domains

Does the Steklov spectrum determine a simply connected strictly convex planar domain with real-analytic boundary up to rigid congruence? Spectral multiplicities are included, and the question remains meaningful for domains smoothly close to a disk.

Why it matters and what counts as progress

Why it matters: Isospectral convex analytic domains would limit what a planar Steklov spectrum reveals even very near a disk.

What counts as progress: Construct a noncongruent isospectral pair in this class or prove spectral rigidity under further geometric assumptions.

How the problem developed

2017 · Open question [1]
Alexandre Girouard and Iosif Polterovich Surveyed the inverse Steklov problem and recorded that no non-isometric Steklov-isospectral Euclidean domains were then known.
11 Aug 2026 · Counterexample [2]
Tao Hu, Jiachen Shi and Quanyu Tang Constructed noncongruent simply connected strictly convex planar domains with real-analytic boundaries and identical Steklov spectra, arbitrarily close to a disk in the smooth topology.

AI contribution

Strictly Convex Steklov-Isospectral Plane Domains

Tao Hu, Jiachen Shi, Quanyu Tang

There are noncongruent simply connected strictly convex Euclidean plane domains with real-analytic boundaries and identical Steklov spectra, including multiplicities. Such pairs can approach a disk smoothly.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull