Dorothee Sch"uth (Humboldt-Universit"at zu Berlin):

"Spectral isolation of bi-invariant metrics on compact Lie groups"

We show that a bi-invariant metric on a compact connected Lie group is
spectrally isolated within the class of left-invariant metrics. In fact,
we prove that given a bi-invariant metric g there exists N such that,
within a neighborhood of g in the class of left-invariant metrics of at
most the same volume, g is uniquely determined by the first N distinct
non-zero eigenvalues of its Laplacian (ignoring multiplicities). In
the case where the Lie group is simple, already N=2 satisfies this property.