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UBC Theses and Dissertations

Deformation of manifolds and continuity of eigenvalues Sun, Yucong


In this thesis, we study the continuity of a family of operators Et acting on functions or sections. First we study two easier cases. 1. The operator is a matrix At; 2. The operator or a Laplacian −Δt which acts on a function. These cases can be solved using min-max techniques coming from linear algebra, which gives us the explicit expression of each eigenvalue. But generally, for elliptic operators Et which acts on a section of a vector bundle, min-max techniques does work. Compared to the Laplacian case, one can not use integration by parts to cancel the Laplacian. And therefore the explicit expression of each eigenvalue can not be obtained. We will introduce Kodaira-Spencer theory for general cases, which is much more powerful and complicated. We’ll introduce the continuity theorem and give detailed proofs for the main theorem.

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