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In this thesis we study the asymptotic behavior of bi-Lipschitz diffeomorphic weighted Riemannian manifolds with techniques from the theory of homogenization. To do so we re-interpret the problem as different induced metrics on one reference manifold. Our analysis is twofold. On the one hand we consider second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our first main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of locally periodic coefficients and study the asymptotic spectral behavior of Euclidean submanifolds with rapidly oscillating geometry. On the other hand we study integral functionals featuring non-convex integrands with non-standard growth on the Euclidean space in a stochastic framework. Our second main result is a Γ-convergence statement under certain assumptions on the statistics of their integrands. Such functionals provide a tool to study the Dirichlet energy on non-uniformly bi-Lipschitz diffeomorphic manifolds. We show Mosco-convergence of the Dirichlet energy and deduce conditions for the spectral behavior of weighted Riemannian manifolds with locally oscillating random structure, especially in the case of Euclidean submanifolds.:Introduction Outline Notation I. Preliminaries 1. Convergence of Riemannian Manifolds 1.1. Hausdorff-Convergence 1.2. Gromov-Hausdorff-Convergence 1.3. Spectral Convergence 1.4. Mosco-Convergence 2. Homogenization 2.1. Periodic Homogenization 2.2. Stochastic Homogenization II. Uniformly bi-Lipschitz Diffeomorphic Manifolds 3. Uniformly Elliptic Operators on a Riemannian Manifold 3.1. Setting 3.2. Main Results 3.3. Strategy of the Proof and Auxiliary Results 3.4. Identi cation of the Limit via Local Coordinate Charts 3.5. Examples 3.6. Proofs 4. Application to Uniformly bi-Lipschitz Diffeomorphic Manifolds 4.1. Setting and Results 4.2. Examples 4.3. Proofs III. Rapidly Oscillating Random Manifolds 5. Integral Functionals with Non-Uniformal Growth 5.1. Setting 5.2. Main Results 5.3. Strategy of the Proof and Auxiliary Results 5.4. Proofs 6. Application to Rapidly Oscillating Riemannian Manifolds 6.1. Setting and Results 6.2. Examples 6.3. Proofs Summary and Discussion Bibliography List of Figures

We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory.

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δg are all simple for a residual set of Cr metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of Cr metrics such that the nonzero eigenvalues of the Hodge Laplacian Δg(k) on k-forms are all simple for 0 ≤ k ≤ 3. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of Cr metrics, the nonzero eigenvalues of the Hodge Laplacian Δg(2) acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *gd, which is related to the Hodge Laplacian by Δg(2) = -(*gd)2 when the operators are restricted to coexact 2-forms on a 5-manifold. We use techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics. We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n = 4 ℓ + 1 and k = 2 ℓ for ℓ ϵ N.

Techn. University, Diss., 2005--Darmstadt.

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in $\R^3$. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced.

Die vorliegende Arbeit beschäftigt sich mit der Höhe (-log Determinante) kompakter nicht-singulärer heisenbergartiger Nilmannigfaltigkeiten. Heisenbergartige Nilmannigfaltigkeiten sind Verallgemeinerungen von Heisenbergmannigfaltigkeiten, d.h., kompakter Quotienten der Heisenberg-Gruppe, ausgestattet mit einer linksinvarianten Metrik. Zunächst werden explizite Formeln für die spektrale Zeta-Funktion und die Höhe bewiesen. Mithilfe dieser Formeln werden im Weiteren mehrere Resultate zur Existenz unterer Schranken/Minima der Höhe auf verschiedenen Moduli bewiesen. Zum Beispiel ist die Höhe stets von unten beschränkt, wenn man nur Metriken vom Heisenberg-Typ und mit Volumen 1 auf einer gegebenen Nilmannigfaltigkeit betrachtet. Im Gegensatz dazu hängt die Existenz unterer Schranken für die Höhe auf dem Modulraum der heisenbergartigen Metriken mit Volumen 1 von der Dimension Modulo 4 der zugrundeliegenden Mannigfaltigkeit ab. Im letzten Abschnitt werden konkrete Minima der Höhe behandelt. Wir zeigen, dass gewisse 3-, 5-, 9- und 25-dimensionale Nilmannigfaltigkeiten vom Heisenberg-Typ lokale Minima sind. Diese stehen in Zusammenhang mit den Minima der Höhe flacher Tori in der jeweiligen Dimension minus 1. Zum Abschluss werden diejenigen linksinvarianten Metriken charakterisiert, an denen die Höhe ein globales Minimum auf einer gegebenen dreidimensionalen Nilmannigfaltigkeit annimmt, indem sie zur Höhe flacher 2-dimensionaler Tori in Bezug gesetzt werden. / This thesis deals with the height (-log determinant) of compact nonsingular Heisenberg-like nilmanifolds. Heisenberg-like nilmanifolds are generalisations of Heisenberg manifolds, i.e., compact quotients of the Heisenberg group endowed with a left invariant metric. First, an explicit formula for the spectral zeta-function and the height is proved. By means of these formulas, several results concerning the existence of lower bounds/minima for the height on different moduli are proved. For example, while the height is always bounded from below when one considers only volume normalised Heisenberg-type metrics on a fixed nilmanifold, the existence of lower bounds for the height on the moduli space of volume normalised Heisenberg-like metrics depends on the dimension modulo 4 of the underlying nilmanifold. In the last part, we consider concrete minima of the height on Heisenberg manifolds. We show that certain 3-, 5-, 9- and 25-dimensional Heisenberg-type nilmanifolds are (local) minima for the height. These nilmanifolds are related to the minima of the height of flat tori in dimensions one less. Finally, the left invariant metrics at which the height attains a global minimum on any three-dimensional nilmanifold are characterised by relating them to the height of flat 2-dimensional tori.