Assume that
is a compact Riemannian manifold of
bounded geometry given by restrictions
on its diameter, Ricci curvature
and injectivity radius. Assume we
are given, with some error, the
first eigenvalues of the Laplacian
on
as well as the corresponding eigenfunctions
restricted on an open set in
.
We then construct a stable approximation to the manifold
.
Namely, we construct a metric space
and a Riemannian manifold which differ,
in a proper sense, just a little from
when the above data are given
with a small error. We give
an explicit
--type
stability estimate on how the constructed manifold and the metric on it depend on the errors in
the given data. Moreover a similar stability estimate is derived for the Gelfand inverse problem.
The proof is based on methods from geometric convergence, a quantitative stability estimate
for the unique continuation and a new version of the geometric boundary control method.