Yes, the each gap survives a small perturbation (even a Hoelder perturbation), see
[here][1]. For smoothness of the Laplacian with respect to the metric see [here][2]. 


  [1]: http://www.mat.univie.ac.at/~michor/hoelder-perturb.pdf
  [2]: http://www.mat.univie.ac.at/~michor/rie-met2.pdf