Boundary Value Problems for Elliptic Differential Operators of First Order

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.Comment: 79 pages, 6 figures, minor corrections, references adde

Eigenvalues and Holonomy

We estimate the eigenvalues of connection Laplacians in terms of the non-triviality of the holonomy.Comment: 9 page

Small eigenvalues of surfaces - old and new

We discuss our recent work on small eigenvalues of surfaces. As an introduction, we present and extend some of the by now classical work of Buser and Randol and explain novel ideas from articles of S\'evennec, Otal, and Otal-Rosas which are of importance in our line of thought.Comment: 24 pages, 5 figures, all comments welcom

On the bottom of spectra under coverings

For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $\Gamma_1\subseteq\Gamma_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $\Gamma_0$ on $\Gamma_1\backslash\Gamma_0$ is amenable.Comment: 8 pages, fixed a technical mistake concerning the volume of the boundary of fundamental domain

On the analytic systole of Riemannian surfaces of finite type

In our previous work we introduced, for a Riemannian surface $S$, the quantity $\Lambda(S):=\inf_F\lambda_0(F)$, where $\lambda_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $\Lambda(S)\ge\lambda_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $\Lambda(S)$ with the systole, improving a result by the last named author.Comment: 35 pages, 1 figure; v2: slightly reorganized, fixed a technical problem in the proof of Thm. 7.3 (v2), added some references, to appear in GAF