  We prove stability estimates for the isoperimetric inequalities for the first
and the second nonzero Laplace eigenvalues on surfaces, both globally and in a
fixed conformal class. We employ the notion of eigenvalues of measures and show
that if a normalized eigenvalue is close to its maximal value, the
corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set
of maximizing measures. In particular, this implies a qualitative stability
result: metrics almost maximizing the normalized eigenvalue must be
$W^{-1,2}$-close to a maximal metric. Following this approach, we prove sharp
quantitative stability of the celebrated Hersch's inequality for the first
eigenvalue on the sphere, as well as of its counterpart for the second
eigenvalue. Similar results are also obtained for the precise isoperimetric
eigenvalue inequalities on the projective plane, torus, and Klein bottle. The
square of the $W^{-1,2}$ distance to a maximizing measure in these stability
estimates is controlled by the difference between the normalized eigenvalue and
its maximal value, indicating that the maxima are in a sense nondegenerate. We
construct examples showing that the power of the distance can not be improved,
and that the choice of the Sobolev space $W^{-1,2}$ is optimal.
