The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an Hermitian or Euclidean metric, and let $g\_u,\: u\in[0,1]$ be a smooth family of Riemannian metrics on $M$. For each $p=0,\ldots,{\rm dim}\: M$ we get a map ${\rm sp}(\Delta^p):\: [0,1] \to {\mathbb R}^{\mathbb N}$ which associates tu $u$ the ordered spectrum (with multiplicities) of the Laplace operator on $p$-forms on $(M,g\_u)$ with coefficients in $E$. The question is: is this map continuous (with respect to the product topology on ${\mathbb R}^{\mathbb N}$)?


I believe that this is true, and that you can prove it using the min-max characterization of the spectrum, but as it seems a rather basic question I would think that somebody already answered some version of it. 



Continuity of the spectrum with respect to the metric
However, for continuity of individual eigenvalues (with appropriate provisos in the presence of multiplicity), the min-max characterization works fine. More refined statements (e.g. about the behavior of the spectrum for smooth or analytic one-parameter families of metrics) are contained in Kato's classic book `Perturbation theory for Linear Operators'. 


