Approximation of Higher Degree Spectra Results for Twisted Laplace Operators
Date
2021
Authors
Fresacher, Matthias Eduard
Editors
Advisors
Leistner, Thomas
Hochs, Peter
Buchdahl, Nicholas
Hochs, Peter
Buchdahl, Nicholas
Journal Title
Journal ISSN
Volume Title
Type:
Thesis
Citation
Statement of Responsibility
Conference Name
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.
School/Discipline
School of Mathematical Sciences
Dissertation Note
Thesis (MPhil) -- University of Adelaide, School of Mathematical Sciences, 2021
Provenance
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