Carolyn Gordon, David L. Webb and Scott Wolpert problem

Question

I didn't find any reference on the subject Isospectral vs Isometry of the problem of Carolyn Gordon, David L. Webb and Scott Wolpert. Could anyone be able give me a book I may consult having a complete answer to their work? Precisely on the following problem :

Thanks in advance!

Answer 1

reference on the subject Isospectral vs Isometry of the problem of Carolyn Gordon, David L. Webb and Scott Wolpert.

The problem is from an article by Marc Kac and is known as "can you hear the shape of a drum". The negative solution in high dimension was found by Milnor before Kac's paper, but getting examples for Kac's problem about planar domains was unsolved until the Gordon-Webb-Wolpert paper. Although the problem is geometric the solution is essentially group theoretic, using Sunada's method.

Rosenberg's book The Laplacian on a Riemannian Manifold has a proof of Sunada's theorem and might be an easier starting point.

https://en.wikipedia.org/wiki/Isospectral

https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

Answer 2

According to MathSciNet those three people have two co-authored papers:

• Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110 (1992), no. 1, 1–22
• One cannot hear the shape of a drum. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 134–138.

A link to the latter paper can be found on this page.