For any bounded regular domain $\Omega$ of a real analytic Riemannian
manifold
$M$ we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the
Dirichlet
Laplacian
of $\Omega$. In this paper, we consider $\lambda_k$ as a
functional on the set of domains of fixed volume in $M$. We
introduce and investigate a natural notion of critical domain for
this functional. In particular, we obtain necessary and
sufficient conditions for a domain to be critical, locally
minimizing or locally maximizing for $\lambda_k$. These
results rely on Hadamard type variational formulae that we
establish in this general setting.
¶
As an application, we obtain a characterization of critical
domains of the trace of the heat kernel under Dirichlet
boundary conditions.