Prof. Dr. Christian Bär

We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary.

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped  with   strictly log-concave warping functions. This generalizes earlier results of  Cecchini-Zeidler to all dimensions. 

Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 minus two antipodal points, thus resolving a problem in Gromov's  ``Four Lectures'' in all dimensions. 

Our arguments are based on spin geometry.

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well under\-stood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.
 

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.

We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We provide examples which are conveniently treated by our methods.

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.
The main application is a general approximation result by sections which have very restrictive local properties an open dense subsets. This shows, for instance, that given any K every manifold of dimension at least two carries a complete C^1,1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course this is impossible for C²-metrics in general.

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n⁴) where n is the size of the matrix. We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold.

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions.
In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.

We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact we do not assume self-adjointness of the Dirac operator on the spacetime or the associated elliptic Dirac operator on the boundary. In this case integration of our local index theorem results in a generalization of previously known index theorems for globally hyperbolic spacetimes that allows for twisting bundles associated with non-compact gauge groups.

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres and 3-dimensional spherical space forms.

Das Werk bietet eine Einführung in die lineare Algebra und die analytische Geometrie und enthält Material für eine zweisemestrige Vorlesung. Es beginnt mit einem Kapitel, das allgemein in die mathematische Denkweise und Beweistechniken einführt, um dann über lineare Gleichungssysteme zur linearen Algebra überzuleiten. Besonderer Wert wird auf eine enge Verzahnung von algebraischen und geometrischen Konzepten gelegt, zum einen um eine gute geometrische Intuition für algebraische Begriffe zu entwickeln, zum anderen um elegante algebraische Beweismethoden für geometrische Sätze einsetzen zu können. Schließlich sind interaktive Übungsseiten und Illustrationen integriert, die zu einem aktiven Studium anregen.

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to CP^2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S^2 x S^2 holds in this class of manifolds.

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined.
We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to H\"ormander.

We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohomology theory and the model given by differential characters. Fiber integration for fibers with boundary is treated in the context of relative differential characters. As applications we treat higher-dimensional holonomy, parallel transport, and transgression.

This text provides a systematic introduction to differential characters, as introduced by Cheeger and Simons. Differential characters form a model of what is nowadays called differential cohomology. In degree 2, integral cohomology of a space X classifies U(1)-bundles over X via the first Chern class, while differential characters correspond to U(1)-bundles with a connection. Similarly, in degree 3, integral cohomology classes classify gerbes over X while differential characters correspond to gerbes with additional geometric structure. We construct the product which provides differential cohomology with a ring structure and we describe the fiber integration map. In both cases, we show uniqueness in the sense that these operations are determined by certain natural axioms. This shows in particular that the various very different descriptions in the literature are equivalent. We present natural and explicit geometric formulas for both the product and the fiber integration map. The underlying space X may be more general than a finite-dimensional manifold. We allow for “smooth spaces” which contains loop spaces of manifolds, for instance. This is important for applications like the transgression map. Up to now, there does not exist much literature on the relative version of differential characters. We investigate them in detail. In degree 2, a relative differential character corresponds to a U(1)-bundle with connection and a section over a subspace. We derive long exact sequences which relate absolute and relative differential characters. Fiber integration for fibers with boundary is naturally considered in the relative framework. The module structure of relative differential cohomology over the ring of absolute differential characters is derived. We discuss various applications including chain field theories and higher dimensional holonomies which occur as actions in string theory.

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which cover all elliptic, overdetermined elliptic, subelliptic and parabolic equations.
We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p-solution must vanish.

We provide a systematic construction of bosonic and fermionic locally covariant quantum field theories on curved backgrounds for large classes of free fields. It turns out that bosonic quantization is possible under much more general assumptions than fermionic quantization.

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p-function.
We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.

This volume contains a collection of surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.

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.

We construct bosonic and fermionic locally covariant quantum field theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.

This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schr\"odinger operators with L^∞-potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.

English translation of the book "Elementare Differentialgeometrie", published by deGruyter.

Dieses Lehrbuch bietet eine Einführung in die Differentialgeometrie von Kurven und Flächen. Es ist in der vorliegenden zweiten, überarbeiteten Auflage um Lösungshinweise sowie Anwendungen in der Kartografie erweitert. Themen sind u. a. euklidische Geometrie, Kurventheorie, Flächentheorie, Krümmungsbegriffe, Minimalflächen, Riemann'sche Geometrie und der Satz von Gauß-Bonnet.

We summarize the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds.

Let H_h = h² L + V where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H_h as h→0. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive h by the classical partition function.

It has been suggested in 1999 that a certain volume growth condition for geodesically complete Riemannian manifolds might imply that the manifold is stochastically complete. This is motivated by a large class of examples and by a known analogous criterion for recurrence of Brownian motion. We show that the suggested implication is not true in general. We also give counter-examples to a converse implication.

This is a survey on the theory of C*-algebras as needed for quantum field theory on curved spacetimes. It covers basic definitions, the spectrum, morphisms, states and representations, product states, and Weyl systems.

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.

After some decades of work a satisfactory theory of quantum gravity is still not available; moreover, there are indications that the original field theoretical approach may be better suited than originally expected. There, to first approximation, one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes. Ingredients of this approach are the formulation of quantum physics in terms of C*-algebras, the geometry of Lorentzian manifolds, in particular their causal structure, and linear hyperbolic differential equations where the well-posedness of the Cauchy problem plays a distinguished role, as well as more recently the insights from suitable concepts such as microlocal analysis. This primer is an outgrowth of a compact course given by the editors and contributing authors to an audience of advanced graduate students and young researchers in the field, and assumes working knowledge of differential geometry and functional analysis on the part of the reader.

Die moderne Theorie der Elementarteilchen, die Quanten-Yang-Mills-Theorie, vermag die experimentellen Befunde mit unerhörter Genauigkeit wiederzugegen, es fehlt ihr jedoch bisher ein solides mathematisches Fundament. Über dessen Gestalt gibt es nur sehr nebelhafte Vorstellungen.

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.

It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It seems to be a folklore fact that the metric can be reconstructed up to conformal equivalence if one replaces the Dirac operator D by sign(D). We give a precise formulation and proof of this fact.

This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail.
The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory.

We give a very condensed introduction to the theory of differential manifolds.

We survey results on the spectrum of the Dirac operator on compact and on complete noncompact Riemannian spin manifolds.

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

We describe the heat kernel asymptotics for roots of a Laplace type operator Δ on a closed manifold. A previously known relation between the Wodzicki residue of Δ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.

The ζ-regularized determinants of the Dirac operator and of its square are computed on spherical space forms. On S² the determinant of Dirac operators twisted by a complex line bundle is also calculated.

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.

Exposition of heat asymptotics for generalized Laplace operators, Weyl's asymptotics for the eigenvalues, variation of the total scalar curvature functional (Einstein-Hilbert action), and of a theorem by Kalau/Walze and Kastler independently relating the second heat coefficient to the Wodzicki residue.

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. The corresponding estimate also holds for the L²-spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in R³.

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order.
The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any L²-section φ contained in a closed A-invariant subspace onto which the restriction of A is semi-bounded has the unique continuation property: if φ vanishes on a non-empty open subset of M, then it vanishes on all of M.

It has recently been conjectured that the eigenvalues λ of the Dirac operator on a closed Riemannian spin manifold M of dimension n ≥ 3 can be estimated from below by the total scalar curvature:

<tex>\lambda^2\geq\frac{n}{4(n-1)}\cdot\frac{\int_{M} S}{vol(M)}</tex>

We show by example that such an estimate is impossible.

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2- or 3-manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak unique continuation property, i.e. if u vanishes on a nonempty open subset of M, then it vanishes on all of M.

The theme is the influence of the spin structure on the Dirac spectrum of a spin manifold. We survey examples and results related to this question.

If $G$ is the structure group of a manifold $M$ it is shown how a certain ideal in the character ring of $G$ corresponds to the set of geometric elliptic operators on $M$. This provides a simple method to construct these operators. For classical structure groups like $G = O(n)$ (Riemannian manifolds), $G = SO(n)$ (oriented Riemannian manifolds), $G = U(m)$ (almost complex manifolds), $G = Spin(n)$ (spin manifolds), or $G = Spin^c(n)$ (spin$^c$ manifolds) this yields well known classical operators like the Euler-deRham operator, signature operator, Cauchy-Riemann operator, or the Dirac operator. For some less well studied structure groups like $Spin^h(n)$ or $Sp(q)Sp(1)$ we can determine the corresponding operators. As applications, we obtain integrality results for such manifolds by applying the Atiyah-Singer Index Theorem to these operators. Finally, we explain how immersions yield interesting structure groups to which one can apply this method. This yields lower bounds on the codimension of immersions in terms of topological data of the manifolds involved.

The Bochner-Lichnerowicz formula and the Atiyah-Singer Index Formula for the Dirac operator have been used to find an obstruction (the $\widehatA$-genus) to producing metrics of positive scalar curvature on spin manifolds. Here the technique is applied to twisted Dirac operators in order to obtain upper bounds on the minimum of the scalar curvature for Riemannian manifolds which admit certain contractive spin mappings into a fixed Riemannian manifold. The principal application is to obtain such upper bounds for algebraic varieties equipped with arbitrary metrics, which admit contractive maps into P^n(C) homotopic to inclusions.

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n-2)-dimensional submanifolds. Hence it is countably (n-2)-rectifiable and its Hausdorff dimension is at most n-2. Moreover, it has locally finite (n-2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n-2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.

This is a survey. We study the question to what extend classical Hodge-deRham theory for harmonic differential forms carries over to harmonic spinors. Despite some special phenomena in very low dimensions and despite the Atiyah-Singer index theorem which provides a link between harmonic spinors and the topology of the underlying manifold it turns out that in many dimensions harmonic spinors are not topologically obstructed. In this respect harmonic spinors behave very differently from harmonic differential forms. We also discuss parallel spinors and Killing spinors

Using Kato's comparison principle for heat semi-groups we derive estimates for the trace of the heat operator on surfaces with variable curvature. This estimate is from above for positively curved surfaces of genus 0 and from below for genus $g \ge 2$. It is shown that the estimates are asymptotically sharp for small time and in the case of positive curvature also for large time. As a consequence we can estimate the corresponding $\zeta$-function by the Riemann $\zeta$-function.

In Kaluza-Klein theory one usually computes the scalar curvature of the principal bundle manifold using the Levi-Civita connection. Here we consider a natural family of invariant connections on a soldered principal bundle which is then parallelizable and hence spinable. This 3-parameter family includes the Levi-Civita connection and the flat connection. By varying the connection instead of merely scaling the metric on the fibers, there is greater independence among the coupling constants in the scalar curvature. In particular, a large cosmological constant can be avoided in spite of tiny fibers.

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a Δ-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal.

We show that for a suitable class of "Dirac-like" operators there holds a Gluing Theorem for connected sums. More precisely, if M₁ and M₂ are closed Riemannian manifolds of dimension n ≥ 3 together with such operators, then the connected sum M₁ # M₂ can be given a Riemannian metric such that the spectrum of its associated operator is close to the disjoint union of the spectra of the two original operators. As an application, we show that in dimension n = 3 mod 4 harmonic spinors for the Dirac operator of a spin, Spin^c, or Spin^h manifold are not topologically obstructed.

The spectrum of the Dirac operator on spherical space forms is calculated. Manifolds with many Killing spinors are characterized. In the last section non-isometric space forms with the same Dirac spectrum are constructed.

We survey relations between the dimension of the solution space of the Dirac equation and the topology of the underlying manifold. It is shown that in certain dimensions existence of metrics with harmonic spinors is not topologically obstructed. In this respect the Dirac operator behaves very differently from the Laplace-Beltrami operator.

We show that every closed spin manifold of dimension n ≡ 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheres Sⁿ, n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sum M₁ # M₂ with certain metrics is close to the union of the spectra of M₁ and of M₂.

Let M be an oriented connected compact Riemannian 4-manifold. We show that if the sectional curvature satisfies K ≥ 1 and the covariant differential of the curvature tensor satisfies |∇R|_∞ ≤ 2/pi, then the intersection form of M is definite.

Let M denote an n-dimensional Riemannian manifold with structure group G. We assume that G acts transitively on the unit sphere in the tangent space and denote by $H \subset G$ the isotropy subgroup for some point in the sphere. We can show Let V_1 and V_2 be two G-Moduls whose restrictions to H are equivalent, then there exists an elliptic pseudo-differential operator from the sections of the bundle associated to V_1 into those for V_2. Consequently, to describe the space of elliptic operators naturally associated to the structure group G we need to determine the kernel R(G,H) of the restriction mapping R(G) -> R(H) of character rings. R(G,H) is a finitely generated ideal in R(G). Here are some examples: i) G = SO(2m), H = SO(2m-1), n=2m, R(G,H) = (1-\Lambda^1\pm\cdots- \Lambda^n-1+1, \Lambda^m_+-\Lambda^m_-). ii) G = Spin(2m), H = Spin(2m-1), R(G,H) = (\Sigma^+-\Sigma^-). iii) G = U(m), H = U(m-1), R(G,H) = (1-\Lambda^1,0\pm\cdots +(-1)^m \Lambda^m,0). We may call the operators corresponding to the generators fundamental. For the above examples this means that for oriented even-dimensional Riemannian manifolds the Euler operator and the signature operator are fundamental, for even-dimensional spin manifolds the Dirac operator is fundamental, and for almost complex manifolds the Cauchy Riemann operator is fundamental. It is interesting to apply this approach to some less well understood structure groups such as G = Sp(q)Sp(1), n=4q. We get elliptic operators for almost quaternionic manifolds and give new integralitiy and vanishing theorems in the case q odd. This method can also be used to study embeddability questions of manifolds into higher dimensional ones.

This is the author's Habilitationsschrift (in German language).

This is the author's dissertation (in German language).