Constructing isospectral manifolds. (English) Zbl 0673.58046

The author gives a lovely account of the construction of isospectral non isometric Riemann surfaces by Buser, the author and Tse using Sunada’s method. He shows there exist Riemann surfaces of genus g for every genus \(g\geq 3\) which are isospectral but not isometric; these surfaces can be taken to have constant negative curvature for \(g\geq 4\). Zoran Luicic and Milica Stojanovic of Belgrade, Yugoslavia have pointed out that there is a mistake in the gluing diagram defining the surfaces of genus 4 which appears on page 833 fig. 2 and fig. 3 of this article; the same mistake appears in figure 4 on page 20 of the article in the Nagoya journal by the author and Tse referred to in the bibliography. All the vertices in these diagrams are glued together giving figures of genus 7 rather than 4.
In a private communication, Brooks indicates the correct glueing diagram should read, moving clockwise from the leftmost edge on the top fundamental domain: B, 1, A, 1, F, 2, C, 4, D, 5, E, 7, E, 4, F, 3, A, 3, B, 2, C, 6, D, 5, G, 7, G, 6. The same change should be made on the other diagram to read A, 1, D, 1, C, 4, B, 2, etc.
Reviewer: P.Gilkey


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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