Spectral rigidity

A class of metrics is spectrally rigid if equality of their specified spectra forces the metrics to be isometric, or, in a deformation version, if continuous isospectral deformations are trivial. The spectrum of a flat torus determines every two-dimensional flat torus up to isometry. The Wolpert generic spectral rigidity theorem is a generic, rather than universal, uniqueness statement.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.