THE ISOSPECTRAL PROBLEM FOR FLAT TORI FROM THREE PERSPECTIVES
Journal article, 2023

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.

Author

Erik Nilsson

Royal Institute of Technology (KTH)

Julie Rowlett

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Felix Rydell

Royal Institute of Technology (KTH)

Bulletin of the American Mathematical Society

0273-0979 (ISSN) 1088-9485 (eISSN)

Vol. 60 1 39-83

Geometric analysis and applications to microbe ecology

Swedish Research Council (VR) (2018-03873), 2019-01-01 -- 2022-12-31.

Subject Categories

Geometry

DOI

10.1090/bull/1770

More information

Latest update

10/27/2023