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In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured Gromov Hausdorff convergence assuming bounds on the sectional curvatures of the manifolds. Later, this result was extended by Cheeger, Colding to only require a lower bound on Ricci Curvature.

Since then, the notion of measured Gromov Hausdorff convergence has been used extensively to study metric measure spaces satisfying for instance the $CD(K,N)$ condition or $RCD(K,N)$ condition.

It is a natural generalization to consider the Laplacian on a Riemannian manifold with boundary assuming some form of boundary condition. For simplicity let's assume Neumann boundary condition.

Is the spectrum of the Laplacian is preserved under measured Gromov Hausdorff convergence for sequences of compact manifolds with boundary under sectional curvature bounds? It seems like this should have been shown somewhere, but I'm not sure if it might be the case that it follows from a more general theorem for instance.

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