Convergence of Riemannian manifolds and spectrum of Laplacian

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Let M_i→M and Y_i→Y, i=1,2,3,..., be two Gromov-Hausdorff convergent sequences of proper metric spaces, where ‘proper' means that any closed bounded set is compact. We give Radon measures on all M_i and M, and assume that the measure on M_i weakly converges to that on M. We are interested in the asymptotic behavior and convergence of maps u_i : M_i→Y_i. We introduce a concept of L^p-convergence of such u_i to a map u : M→Y,p【greater than or equal】1, and establish a theory of convergence of energy functionals E_i defined on the mapping space {u : M_i→Y_i} by generalizing Mosco's variational convergences. Mosco defined the asymptotically compactness of {E_i}, as a generalization of the Rellich compactness. The asymptotic compactness is useful to obtain the convergence of energy minimizers, i.e., harmonic maps. Under a uniform bound of the Poincare constant for E_i and some condition on the metric structure of M, we prove the asymptotic compactness of {E_i}. We say that E_i compactly converges to a functional E on {u : M→Y} if E_i Γ-converges to E and if {E_i} is asymptotically compact. We prove that the compact convergence E_i→E is equivalent to the Gromov-Hausdorff convergence of the E_i-sublevel sets to the E-sublevel sets. This gives a geometric interpretation of the compact convergence. Assume in addition that Y_i are all CAT(0)-spaces and E_i are convex and lower semi-continuous. Then, we prove that the compact convergence E_i→E is equivalent to the convergence of the corresponding resolvents, where the resolvents for E_i and E are defined by using the minimizers of the Moreau-Yosida approximation. As applications, we investigate the spectra of the Korevaar-Schoen approximating energy forms. We also obtain the compactness of the energy functionals over Riemannian manifolds under a bound of Ricci curvature.