Topics Related To The Convergence Of Manifolds

Cheeger and Gromov developed the theory of convergence of Riemannian man-ifolds in the1970s. Now this theory has become an important tool in diferential ge-ometry and has many applications in other theories such as the Ricci flow. Spectraltheory is another important theory in diferential geometry. It plays an important rolein both mathematics and physics. The eigenvalues of Laplace operator of a Riemannianmanifold are closely related to the geometric properties of manifolds. In this thesis wediscuss the convergence of eigenvalues of manifolds which converge in the Cheeger-Gromov sense. We prove that there exists a subsequence whose eigenvalues convergeto eigenvalues of the limit manifolds.Schur Lemma is a fundamental lemma in Riemannian Geometry. This lemmastates that every Einstein manifold has constant scale curvature. Camillo de Lellis andPeter M. Topping point out that on a manifold with nonnegative Ricci curvature if thetraceless Ricci tensor is small rather than identically zero, then the scale curvature isalmost a constant. By means of the theory of submanifolds, we generalize their theoremto manifolds with special boundaries.