Approximation of Higher Degree Spectra Results for Twisted Laplace Operators

Abstract

This thesis examines the eigenvalues of the connection Laplacian acting on differential forms with values in a Hermitian vector bundle with connection over a closed Riemannian manifold. Specifically, building upon previous work by Whitney, Dodziuk, Patodi and Zahariev, a combinatorial analogue of the connection Laplacian is defined via triangulations of the manifold whereby differential forms are associated to cochains. Using the min-max principle as a key ingredient, this reduces the infinite dimensional analytic eigenvalue problem to a finite dimensional combinatorial one. In theory, this allows the eigenvalues to be calculated with numerical methods and sufficient computational power. In this thesis, I prove that the eigenvalues of the analytic Laplacian are bounded below by the eigenvalues of the combinatorial Laplacian for differential forms and cochains of arbitrary degree with values in a trivial complex line bundle provided an assumption is met. This is achieved via an explicit calculation of the growth rate of the Whitney map under standard subdivisions.