# Prof. Dr. Christian Bär

### Professor

Kontakt
Raum:
2.09.0.18
Telefon:
+49 331 977-1348

#### Sprechstunde

findet wegen der aktuellen Corona-Krise bis auf weiteres nicht statt. Sie können mir Ihr Anliegen gerne per E-Mail übermitteln.

#### Arbeitsgebiete

• Differentialgeometrie
• Spektralgeometrie
• globale Analysis
• Anwendungen in der mathematischen Physik

Für Details siehe mein ResearchGate-Profil.

#### Drittmittelprojekte

Meine Arbeitsgruppe ist an folgenden durch Drittmittel finanzierten Projekten beteiligt:

#### Publikationen

2022 | Boundary value problems for general first-order elliptic differential operators | Christian Bär, Lashi BandaraZeitschrift: J. Funct. AnalysisSeiten: 109445Band: 282Link zur Publikation , Link zum Preprint ,

Video abstract: https://vimeo.com/523211595

### Boundary value problems for general first-order elliptic differential operators

#### Autoren: Christian Bär, Lashi Bandara (2022)

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.

Zeitschrift:
J. Funct. Analysis
Seiten:
109445
Band:
282

2022 | Local Flexibility for Open Partial Differential Relations | Christian Bär, Bernhard HankeZeitschrift: Communications on Pure and Applied MathematicsSeiten: 1377-1415Band: 75Link zur Publikation , Link zum Preprint

### Local Flexibility for Open Partial Differential Relations

#### Autoren: Christian Bär, Bernhard Hanke (2022)

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 C1,1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course this is impossible for C2-metrics in general.

Zeitschrift:
Communications on Pure and Applied Mathematics
Seiten:
1377-1415
Band:
75

2021 | The Faddeev-LeVerrier algorithm and the Pfaffian | Christian BärZeitschrift: Linear Algebra and its ApplicationsSeiten: 39-55Band: 630Link zur Publikation , Link zum Preprint

### The Faddeev-LeVerrier algorithm and the Pfaffian

#### Autoren: Christian Bär (2021)

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.

Zeitschrift:
Linear Algebra and its Applications
Seiten:
39-55
Band:
630

2021 | Manifolds with many Rarita-Schwinger fields | Christian Bär, Rafe MazzeoZeitschrift: Commun. Math. Phys.Verlag: Springer-VerlagLink zur Publikation , Link zum Preprint

### Manifolds with many Rarita-Schwinger fields

#### Autoren: Christian Bär, Rafe Mazzeo (2021)

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.

Zeitschrift:
Commun. Math. Phys.
Verlag:
Springer-Verlag

2020 | Boundary conditions for scalar curvature | Christian Bär and Bernhard HankeLink zum Preprint

### Boundary conditions for scalar curvature

#### Autoren: Christian Bär and Bernhard Hanke (2020)

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.

2020 | Local Index Theory for Lorentzian Manifolds | Christian Bär and Alexander StrohmaierLink zum Preprint

### Local Index Theory for Lorentzian Manifolds

#### Autoren: Christian Bär and Alexander Strohmaier (2020)

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.

2019 | An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary | Christian Bär, Alexander StrohmaierZeitschrift: Amer. J. Math.Verlag: Johns Hopkins Univ. PressSeiten: 1421-1455Band: 141 (5)Link zur Publikation , Link zum Preprint

### An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

#### Autoren: Christian Bär, Alexander Strohmaier (2019)

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.

Zeitschrift:
Amer. J. Math.
Verlag:
Johns Hopkins Univ. Press
Seiten:
1421-1455
Band:
141 (5)

2019 | The curl operator on odd-dimensional manifolds | Christian BärZeitschrift: J. Math. Phys.Seiten: 031501Band: 60Link zur Publikation , Link zum Preprint

### The curl operator on odd-dimensional manifolds

#### Autoren: Christian Bär (2019)

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.

Zeitschrift:
J. Math. Phys.
Seiten:
031501
Band:
60

2018 | Lineare Algebra und analytische Geometrie | Christian BärVerlag: SpringerLink zur Publikation

### Lineare Algebra und analytische Geometrie

#### Autoren: Christian Bär (2018)

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.

Verlag:
Springer

2018 | Boundary value problems for the Lorentzian Dirac operator | Christian Bär, Sebastian HannesVerlag: Oxford Univ. PressBuchtitel: A. Dancer, J.E. Andersen, O. García-Prada (eds.): Geometry and PhysicsSeiten: 3-18Band: 1Link zur Publikation , Link zum Preprint

### Boundary value problems for the Lorentzian Dirac operator

#### Autoren: Christian Bär, Sebastian Hannes (2018)

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.

Verlag:
Oxford Univ. Press
Buchtitel:
A. Dancer, J.E. Andersen, O. García-Prada (eds.): Geometry and Physics
Seiten:
3-18
Band:
1

2018 | Wave and Dirac equations on manifolds | Lars Andersson, Christian BärVerlag: de GruyterBuchtitel: Space - Time - MatterSeiten: 324-348Link zur Publikation , Link zum Preprint

### Wave and Dirac equations on manifolds

#### Autoren: Lars Andersson, Christian Bär (2018)

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.

Verlag:
de Gruyter
Buchtitel:
Space - Time - Matter
Seiten:
324-348

2016 | A rigorous geometric derivation of the chiral anomaly in curved backgrounds | Christian Bär, Alexander StrohmaierZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 703-721Band: 347, no. 3Link zur Publikation , Link zum Preprint

### A rigorous geometric derivation of the chiral anomaly in curved backgrounds

#### Autoren: Christian Bär, Alexander Strohmaier (2016)

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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
703-721
Band:
347, no. 3

2016 | Guide to Boundary Value Problems for Dirac-Type Operators | Werner Ballmann, Christian BärReihe: Progress in MathematicsVerlag: Springer-VerlagBuchtitel: Arbeitstagung Bonn 2013Seiten: 43-80Band: 319Link zur Publikation , Link zum Preprint

### Guide to Boundary Value Problems for Dirac-Type Operators

#### Autoren: Werner Ballmann, Christian Bär (2016)

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.

Reihe:
Progress in Mathematics
Verlag:
Springer-Verlag
Buchtitel:
Arbeitstagung Bonn 2013
Seiten:
43-80
Band:
319

2015 | Geometrically formal 4-manifolds with nonnegative sectional curvature | Christian BärZeitschrift: Comm. Anal. Geom.Verlag: International PressSeiten: 479-497Band: 23, no. 3Link zur Publikation , Link zum Preprint

### Geometrically formal 4-manifolds with nonnegative sectional curvature

#### Autoren: Christian Bär (2015)

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.

Zeitschrift:
Comm. Anal. Geom.
Verlag:
International Press
Seiten:
479-497
Band:
23, no. 3

2015 | Green-hyperbolic operators on globally hyperbolic spacetimes | Christian BärZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 1585-1615Band: 333, no. 3Link zur Publikation , Link zum Preprint

### Green-hyperbolic operators on globally hyperbolic spacetimes

#### Autoren: Christian Bär (2015)

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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
1585-1615
Band:
333, no. 3

2015 | Initial value problems for wave equations on manifolds | Christian Bär, Roger Tagne WafoZeitschrift: Math. Phys. Anal. Geom.Verlag: SpringerSeiten: Art.:7Band: 18, no. 1Link zur Publikation , Link zum Preprint

### Initial value problems for wave equations on manifolds

#### Autoren: Christian Bär, Roger Tagne Wafo (2015)

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.

Zeitschrift:
Math. Phys. Anal. Geom.
Verlag:
Springer
Seiten:
Art.:7
Band:
18, no. 1

2014 | Differential Characters and Geometric Chains | Christian Bär, Christian BeckerReihe: Lecture Notes in MathematicsVerlag: SpringerBuchtitel: C. Bär, C. Becker: Differential CharactersSeiten: 1-90Band: 2112Link zur Publikation , Link zum Preprint

### Differential Characters and Geometric Chains

#### Autoren: Christian Bär, Christian Becker (2014)

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.

Reihe:
Lecture Notes in Mathematics
Verlag:
Springer
Buchtitel:
C. Bär, C. Becker: Differential Characters
Seiten:
1-90
Band:
2112

2014 | Differential Characters | Christian Bär, Christian BeckerReihe: Lecture Notes in MathematicsVerlag: SpringerBand: 2112Link zur Publikation

### Differential Characters

#### Autoren: Christian Bär, Christian Becker (2014)

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.

Reihe:
Lecture Notes in Mathematics
Verlag:
Springer
Band:
2112

2013 | Some properties of solutions to weakly hypoelliptic equations | Christian BärZeitschrift: Int. J. Differ. Equ.Verlag: Hindawi, New YorkSeiten: Art. ID 526390Band: 2013Link zur Publikation , Link zum Preprint

### Some properties of solutions to weakly hypoelliptic equations

#### Autoren: Christian Bär (2013)

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.

Zeitschrift:
Int. J. Differ. Equ.
Verlag:
Hindawi, New York
Seiten:
Art. ID 526390
Band:
2013

2012 | CCR- versus CAR-quantization on curved spacetimes | Christian Bär, Nicolas GinouxVerlag: BirkhäuserBuchtitel: F. Finster, O. Müller, M. Nardmann, J. Tolksdorf, E. Zeidler (Eds.): Quantum Field Theory and GravitySeiten: 183-206Link zur Publikation , Link zum Preprint

### CCR- versus CAR-quantization on curved spacetimes

#### Autoren: Christian Bär, Nicolas Ginoux (2012)

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.

Verlag:
Birkhäuser
Buchtitel:
F. Finster, O. Müller, M. Nardmann, J. Tolksdorf, E. Zeidler (Eds.): Quantum Field Theory and Gravity
Seiten:
183-206

2012 | Renormalized Integrals and a Path Integral Formula for the Heat Kernel on a Manifold | Christian BärZeitschrift: Contemp. Math.Verlag: American Mathematical SocietyBuchtitel: C. Aldana, M. Braverman, B. Iochum, C. Neira Jimenez (eds.): Analysis, Geometry and Quantum Field TheorySeiten: 179-197Band: 584Link zur Publikation , Link zum Preprint

### Renormalized Integrals and a Path Integral Formula for the Heat Kernel on a Manifold

#### Autoren: Christian Bär (2012)

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 Lp-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.

Zeitschrift:
Contemp. Math.
Verlag:
American Mathematical Society
Buchtitel:
C. Aldana, M. Braverman, B. Iochum, C. Neira Jimenez (eds.): Analysis, Geometry and Quantum Field Theory
Seiten:
179-197
Band:
584

2012 | Global Differential Geometry | Christian Bär, Joachim Lohkamp, Matthias SchwarzReihe: Springer Proceedings in MathematicsVerlag: SpringerBand: 17Link zur Publikation

### Global Differential Geometry

#### Autoren: Christian Bär, Joachim Lohkamp, Matthias Schwarz (2012)

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.

Reihe:
Springer Proceedings in Mathematics
Verlag:
Springer
Band:
17

2012 | Boundary Value Problems for Elliptic Differential Operators of First Order | Werner Ballmann, Christian BärZeitschrift: Surv. Differ. Geom.Verlag: International PressSeiten: 1-78Band: 17Link zur Publikation , Link zum Preprint

### Boundary Value Problems for Elliptic Differential Operators of First Order

#### Autoren: Werner Ballmann, Christian Bär (2012)

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.

Zeitschrift:
Surv. Differ. Geom.
Verlag:
International Press
Seiten:
1-78
Band:
17

2012 | Classical and Quantum Fields on Lorentzian Manifolds | Christian Bär, Nicolas GinouxReihe: Springer Proceedings in MathematicsVerlag: SpringerBuchtitel: C. Bär, J. Lohkamp, M. Schwarz (Eds.): Global Differential GeometrySeiten: 359-400Band: 17Link zur Publikation , Link zum Preprint

### Classical and Quantum Fields on Lorentzian Manifolds

#### Autoren: Christian Bär, Nicolas Ginoux (2012)

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.

Reihe:
Springer Proceedings in Mathematics
Verlag:
Springer
Buchtitel:
C. Bär, J. Lohkamp, M. Schwarz (Eds.): Global Differential Geometry
Seiten:
359-400
Band:
17

2011 | Wiener Measures on Riemannian Manifolds and the Feynman-Kac Formula | Christian Bär, Frank PfäffleZeitschrift: Mat. Contemp.Verlag: Sociedade Brasileira de MatematicaSeiten: 37-90Band: 40Link zur Publikation , Link zum Preprint

### Wiener Measures on Riemannian Manifolds and the Feynman-Kac Formula

#### Autoren: Christian Bär, Frank Pfäffle (2011)

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.

Zeitschrift:
Mat. Contemp.
Verlag:
Seiten:
37-90
Band:
40

2010 | Elementary Differential Geometry (Monografie) | Christian BärReihe: Cambridge University PressVerlag: Cambridge University PressLink zur Publikation

### Elementary Differential Geometry (Monografie)

#### Autoren: Christian Bär (2010)

Reihe:
Cambridge University Press
Verlag:
Cambridge University Press

2010 | Elementare Differentialgeometrie (Monografie), 2. Auflage | Christian BärReihe: de Gruyter LehrbuchVerlag: De GruyterSeiten: xvi+340 pagesLink zur Publikation

### Elementare Differentialgeometrie (Monografie), 2. Auflage

#### Autoren: Christian Bär (2010)

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.

Reihe:
de Gruyter Lehrbuch
Verlag:
De Gruyter
Seiten:
xvi+340 pages

2010 | Linear Wave Equations on Lorentzian Manifolds | Christian BärVerlag: European Mathematical SocietyBuchtitel: V. Cortes (ed.): Handbook of Pseudo-Riemannian Geometry and SupersymmetrySeiten: 897-914Link zur Publikation , Link zum Preprint

### Linear Wave Equations on Lorentzian Manifolds

#### Autoren: Christian Bär (2010)

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

Verlag:
European Mathematical Society
Buchtitel:
V. Cortes (ed.): Handbook of Pseudo-Riemannian Geometry and Supersymmetry
Seiten:
897-914

2010 | Asymptotic heat kernel expansion in the semi-classical limit | Christian Bär, Frank PfäffleZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 731-744Band: 294, no. 3Link zur Publikation , Link zum Preprint

### Asymptotic heat kernel expansion in the semi-classical limit

#### Autoren: Christian Bär, Frank Pfäffle (2010)

Let Hh = 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 Hh 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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
731-744
Band:
294, no. 3

2010 | Stochastic completeness and volume growth | Christian Bär, Gregorio Pacelli BessaZeitschrift: Proc. Amer. Math. Soc.Verlag: American Mathematical SocietySeiten: 2629-2640Band: 138, no. 7Link zur Publikation , Link zum Preprint

### Stochastic completeness and volume growth

#### Autoren: Christian Bär, Gregorio Pacelli Bessa (2010)

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.

Zeitschrift:
Proc. Amer. Math. Soc.
Verlag:
American Mathematical Society
Seiten:
2629-2640
Band:
138, no. 7

2009 | C*-algebras | Christian Bär, Christian BeckerReihe: Lecture Notes in PhysicsVerlag: SpringerBuchtitel: C. Bär, K. Fredenhagen (eds.): Quantum Field Theory on Curved SpacetimesSeiten: 1-37Band: 786Link zur Publikation , Link zum Preprint

### C*-algebras

#### Autoren: Christian Bär, Christian Becker (2009)

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.

Reihe:
Lecture Notes in Physics
Verlag:
Springer
Buchtitel:
C. Bär, K. Fredenhagen (eds.): Quantum Field Theory on Curved Spacetimes
Seiten:
1-37
Band:
786

2009 | Spectral Bounds for Dirac Operators on Open Manifolds | Christian BärZeitschrift: Ann. Global Anal. Geom.Verlag: SpringerSeiten: 67-79Band: 36, no. 1Link zur Publikation , Link zum Preprint

### Spectral Bounds for Dirac Operators on Open Manifolds

#### Autoren: Christian Bär (2009)

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.

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
67-79
Band:
36, no. 1

2009 | Quantum Field Theory on Curved Spacetimes - Concepts and Mathematical Foundations (Monografie) | Christian Bär, Klaus FredenhagenReihe: Lecture Notes in PhysicsVerlag: SpringerBand: 786Link zur Publikation

### Quantum Field Theory on Curved Spacetimes - Concepts and Mathematical Foundations (Monografie)

#### Autoren: Christian Bär, Klaus Fredenhagen (2009)

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.

Reihe:
Lecture Notes in Physics
Verlag:
Springer
Band:
786

2009 | Das Yang-Mills-Problem. Die mathematische Zähmung des Standardmodells | Christian Bär, Christoph StephanZeitschrift: Spektrum der WissenschaftVerlag: Spektrum der Wissenschaft VerlagSeiten: 66-73Band: 5Link zur Publikation

### Das Yang-Mills-Problem. Die mathematische Zähmung des Standardmodells

#### Autoren: Christian Bär, Christoph Stephan (2009)

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.

Zeitschrift:
Spektrum der Wissenschaft
Verlag:
Spektrum der Wissenschaft Verlag
Seiten:
66-73
Band:
5

2008 | Path integrals on manifolds by finite dimensional approximation | Christian Bär, Frank PfäffleZeitschrift: J. Reine Angew. Math.Verlag: De GruyterSeiten: 29-57Band: 625Link zur Publikation , Link zum Preprint

### Path integrals on manifolds by finite dimensional approximation

#### Autoren: Christian Bär, Frank Pfäffle (2008)

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.

Zeitschrift:
J. Reine Angew. Math.
Verlag:
De Gruyter
Seiten:
29-57
Band:
625

2007 | Conformal structures in noncommutative geometry | Christian BärZeitschrift: J. Noncommut. Geom.Verlag: European Mathematical SocietySeiten: 385-395Band: 1, no. 3Link zur Publikation , Link zum Preprint

### Conformal structures in noncommutative geometry

#### Autoren: Christian Bär (2007)

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.

Zeitschrift:
J. Noncommut. Geom.
Verlag:
European Mathematical Society
Seiten:
385-395
Band:
1, no. 3

2007 | Wave equations on Lorentzian manifolds and quantization (Monografie) | Christian Bär, Nicolas Ginoux, Frank PfäffleReihe: ESI Lectures in Mathematics and PhysicsVerlag: European Mathematical SocietySeiten: viii+194 pagesLink zur Publikation , Link zum Preprint

### Wave equations on Lorentzian manifolds and quantization (Monografie)

#### Autoren: Christian Bär, Nicolas Ginoux, Frank Pfäffle (2007)

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.

Reihe:
ESI Lectures in Mathematics and Physics
Verlag:
European Mathematical Society
Seiten:
viii+194 pages

2005 | Introduction to differential manifolds | Christian BärVerlag: International PressBuchtitel: J.-P. Bourguignon et al. (eds.): Dirac operators: yesterday and todaySeiten: 13-25Link zur Publikation , Link zum Preprint

### Introduction to differential manifolds

#### Autoren: Christian Bär (2005)

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

Verlag:
International Press
Buchtitel:
J.-P. Bourguignon et al. (eds.): Dirac operators: yesterday and today
Seiten:
13-25

2005 | The spectrum of the Dirac operator | Christian BärVerlag: International PressBuchtitel: J.-P. Bourguignon et al. (eds.): Dirac operators: yesterday and todaySeiten: 145-162Link zur Publikation , Link zum Preprint

### The spectrum of the Dirac operator

#### Autoren: Christian Bär (2005)

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

Verlag:
International Press
Buchtitel:
J.-P. Bourguignon et al. (eds.): Dirac operators: yesterday and today
Seiten:
145-162

2005 | Generalized cylinders in semi-Riemannian and Spin geometry | Christian Bär, Paul Gauduchon, Andrei MoroianuZeitschrift: Math. Z.Verlag: SpringerSeiten: 545-580Band: 249, no. 3Link zur Publikation , Link zum Preprint

### Generalized cylinders in semi-Riemannian and Spin geometry

#### Autoren: Christian Bär, Paul Gauduchon, Andrei Moroianu (2005)

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.

Zeitschrift:
Math. Z.
Verlag:
Springer
Seiten:
545-580
Band:
249, no. 3

2004 | The first Dirac eigenvalues on manifolds with positive scalar curvature | Christian Bär, Mattias DahlZeitschrift: Proc. Amer. Math. Soc.Verlag: American Mathematical SocietySeiten: 3337-3344 (electronic)Band: 132, no. 1Link zur Publikation , Link zum Preprint

### The first Dirac eigenvalues on manifolds with positive scalar curvature

#### Autoren: Christian Bär, Mattias Dahl (2004)

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.

Zeitschrift:
Proc. Amer. Math. Soc.
Verlag:
American Mathematical Society
Seiten:
3337-3344 (electronic)
Band:
132, no. 1

2003 | Heat kernel asymptotics for roots of generalized Laplacians | Christian Bär, Sergiu MoroianuZeitschrift: Internat. J. Math.Verlag: World ScientificSeiten: 397-412Band: 14, no. 4Link zur Publikation , Link zum Preprint

### Heat kernel asymptotics for roots of generalized Laplacians

#### Autoren: Christian Bär, Sergiu Moroianu (2003)

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.

Zeitschrift:
Internat. J. Math.
Verlag:
World Scientific
Seiten:
397-412
Band:
14, no. 4

2003 | The Dirac determinant of spherical space forms | Christian Bär, Sven SchopkaVerlag: SpringerBuchtitel: S. Hildebrandt, H. Karcher (Eds.): Geometric analysis and nonlinear partial differential equationsSeiten: 39-67Link zur Publikation , Link zum Preprint

### The Dirac determinant of spherical space forms

#### Autoren: Christian Bär, Sven Schopka (2003)

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.

Verlag:
Springer
Buchtitel:
S. Hildebrandt, H. Karcher (Eds.): Geometric analysis and nonlinear partial differential equations
Seiten:
39-67

2003 | Small eigenvalues of the conformal Laplacian | Christian Bär, Mattias DahlZeitschrift: Geom. Funct. Anal.Verlag: SpringerSeiten: 483-508Band: 13, no. 3Link zur Publikation , Link zum Preprint

### Small eigenvalues of the conformal Laplacian

#### Autoren: Christian Bär, Mattias Dahl (2003)

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.

Zeitschrift:
Geom. Funct. Anal.
Verlag:
Springer
Seiten:
483-508
Band:
13, no. 3

2002 | The Einstein-Hilbert action as a spectral action | Bernd Ammann, Christian BärReihe: Lecture Notes in PhysicsVerlag: SpringerBuchtitel: F. Scheck, H. Upmeier, W. Werner (eds.): Noncommutative geometry and the standard model of elementary particle physics (Hesselberg, 1999)Seiten: 75-108Band: 596Link zur Publikation , Link zum Preprint

### The Einstein-Hilbert action as a spectral action

#### Autoren: Bernd Ammann, Christian Bär (2002)

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.

Reihe:
Lecture Notes in Physics
Verlag:
Springer
Buchtitel:
F. Scheck, H. Upmeier, W. Werner (eds.): Noncommutative geometry and the standard model of elementary particle physics (Hesselberg, 1999)
Seiten:
75-108
Band:
596

2002 | Dirac eigenvalue estimates on surfaces | Bernd Ammann, Christian BärZeitschrift: Math. Z.Verlag: SpringerSeiten: 423-449Band: 240, no. 2Link zur Publikation , Link zum Preprint

### Dirac eigenvalue estimates on surfaces

#### Autoren: Bernd Ammann, Christian Bär (2002)

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.

Zeitschrift:
Math. Z.
Verlag:
Springer
Seiten:
423-449
Band:
240, no. 2

2002 | Surgery and the Spectrum of the Dirac Operator | Christian Bär, Mattias DahlZeitschrift: J. Reine Angew. Math.Verlag: De GruyterSeiten: 53-76Band: 552Link zur Publikation , Link zum Preprint

### Surgery and the Spectrum of the Dirac Operator

#### Autoren: Christian Bär, Mattias Dahl (2002)

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.

Zeitschrift:
J. Reine Angew. Math.
Verlag:
De Gruyter
Seiten:
53-76
Band:
552

2001 | Semi-Bounded Restrictions of Dirac Type Operators and the Unique Continuation Property | Christian Bär, Alexander StrohmaierZeitschrift: Differential Geom. Appl.Verlag: ElsevierSeiten: 175-182Band: 15, no. 2Link zur Publikation , Link zum Preprint

### Semi-Bounded Restrictions of Dirac Type Operators and the Unique Continuation Property

#### Autoren: Christian Bär, Alexander Strohmaier (2001)

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.

Zeitschrift:
Differential Geom. Appl.
Verlag:
Elsevier
Seiten:
175-182
Band:
15, no. 2

2001 | Elementare Differentialgeometrie (Monografie) | Christian BärReihe: de Gruyter LehrbuchVerlag: De GruyterSeiten: xii+281 pagesLink zur Publikation

### Elementare Differentialgeometrie (Monografie)

#### Autoren: Christian Bär (2001)

Reihe:
de Gruyter Lehrbuch
Verlag:
De Gruyter
Seiten:
xii+281 pages

2000 | Dirac eigenvalues and total scalar curvature | Bernd Ammann, Christian BärZeitschrift: J. Geom. Phys.Verlag: ElsevierSeiten: 229-234Band: 33, no.3-4Link zur Publikation , Link zum Preprint

### Dirac eigenvalues and total scalar curvature

#### Autoren: Bernd Ammann, Christian Bär (2000)

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.

Zeitschrift:
J. Geom. Phys.
Verlag:
Elsevier
Seiten:
229-234
Band:
33, no.3-4

2000 | The Dirac Operator on Hyperbolic Manifolds of Finite Volume | Christian BärZeitschrift: J. Differential Geom.Seiten: 439-488Band: 54, no. 3Link zur Publikation , Link zum Preprint

### The Dirac Operator on Hyperbolic Manifolds of Finite Volume

#### Autoren: Christian Bär (2000)

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.

Zeitschrift:
J. Differential Geom.
Seiten:
439-488
Band:
54, no. 3

2000 | Localization and Semibounded Energy - A Weak Unique Continuation Theorem | Christian BärZeitschrift: J. Geom. Phys.Verlag: ElsevierSeiten: 155-161Band: 34, no. 2Link zur Publikation , Link zum Preprint

### Localization and Semibounded Energy - A Weak Unique Continuation Theorem

#### Autoren: Christian Bär (2000)

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.

Zeitschrift:
J. Geom. Phys.
Verlag:
Elsevier
Seiten:
155-161
Band:
34, no. 2

2000 | Dependence of the Dirac spectrum on the spin structure | Christian BärZeitschrift: Semin. Congr.Verlag: Societe Mathematique de FranceBuchtitel: Global Analysis and Harmonic AnalysisSeiten: 17-33Band: 4Link zur Publikation , Link zum Preprint

### Dependence of the Dirac spectrum on the spin structure

#### Autoren: Christian Bär (2000)

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.

Zeitschrift:
Semin. Congr.
Verlag:
Societe Mathematique de France
Buchtitel:
Global Analysis and Harmonic Analysis
Seiten:
17-33
Band:
4

1999 | Elliptic Symbols | Christian BärZeitschrift: Math. Nachr.Verlag: Wiley-VCHSeiten: 7-35Band: 201, no. 1Link zur Publikation , Link zum Preprint

### Elliptic Symbols

#### Autoren: Christian Bär (1999)

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.

Zeitschrift:
Math. Nachr.
Verlag:
Wiley-VCH
Seiten:
7-35
Band:
201, no. 1

1999 | The Dirac Operator and the Scalar Curvature of Continuously Deformed Algebraic Varieties | Christian Bär, David BleeckerZeitschrift: Contemp. Math.Verlag: American Mathematical SocietySeiten: 3-24Band: 242Link zur Publikation , Link zum Preprint

### The Dirac Operator and the Scalar Curvature of Continuously Deformed Algebraic Varieties

#### Autoren: Christian Bär, David Bleecker (1999)

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.

Zeitschrift:
Contemp. Math.
Verlag:
American Mathematical Society
Seiten:
3-24
Band:
242

1999 | Zero Sets of Solutions to Semilinear Elliptic Systems of First Order | Christian BärZeitschrift: Invent. Math.Verlag: SpringerSeiten: 183-202Band: 138, no. 1Link zur Publikation , Link zum Preprint

### Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

#### Autoren: Christian Bär (1999)

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.

Zeitschrift:
Invent. Math.
Verlag:
Springer
Seiten:
183-202
Band:
138, no. 1

1998 | On Harmonic Spinors | Christian BärZeitschrift: Acta Phys. Polon. BVerlag: Jagiellonian University, PolandSeiten: 859-869Band: 29, no. 4Link zur Publikation , Link zum Preprint

### On Harmonic Spinors

#### Autoren: Christian Bär (1998)

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

Zeitschrift:
Acta Phys. Polon. B
Verlag:
Jagiellonian University, Poland
Seiten:
859-869
Band:
29, no. 4

1998 | Heat Operator and Zeta-Function Estimates for Surfaces | Christian BärZeitschrift: Arch. Math.Verlag: SpringerSeiten: 63-70Band: 71, no. 1Link zur Publikation , Link zum Preprint

### Heat Operator and Zeta-Function Estimates for Surfaces

#### Autoren: Christian Bär (1998)

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.

Zeitschrift:
Arch. Math.
Verlag:
Springer
Seiten:
63-70
Band:
71, no. 1

1998 | Connections on Soldered Principal Bundles | Christian Bär, David BleeckerZeitschrift: Acta Phys. Polon. BVerlag: Jagiellonian University, PolandSeiten: 891-903Band: 29, no. 4Link zur Publikation , Link zum Preprint

### Connections on Soldered Principal Bundles

#### Autoren: Christian Bär, David Bleecker (1998)

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.

Zeitschrift:
Acta Phys. Polon. B
Verlag:
Jagiellonian University, Poland
Seiten:
891-903
Band:
29, no. 4

1998 | The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles | Bernd Ammann, Christian BärZeitschrift: Ann. Global Anal. Geom.Verlag: SpringerSeiten: 221-253Band: 16, no. 3Link zur Publikation , Link zum Preprint

### The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

#### Autoren: Bernd Ammann, Christian Bär (1998)

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.

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
221-253
Band:
16, no. 3

1998 | Extrinsic Bounds for Eigenvalues of the Dirac Operator | Christian BärZeitschrift: Ann. Global Anal. Geom.Verlag: SpringerSeiten: 573-596Band: 16, no. 6Link zur Publikation , Link zum Preprint

### Extrinsic Bounds for Eigenvalues of the Dirac Operator

#### Autoren: Christian Bär (1998)

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.

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
573-596
Band:
16, no. 6

1997 | On Nodal Sets for Dirac and Laplace Operators | Christian BärZeitschrift: Comm. Math. Phys.Verlag: SpringerSeiten: 709-721Band: 188, no. 3Link zur Publikation , Link zum Preprint

### On Nodal Sets for Dirac and Laplace Operators

#### Autoren: Christian Bär (1997)

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.

Zeitschrift:
Comm. Math. Phys.
Verlag:
Springer
Seiten:
709-721
Band:
188, no. 3

1997 | Harmonic Spinors for Twisted Dirac Operators | Christian BärZeitschrift: Math. Ann.Verlag: SpringerSeiten: 225-246Band: 309, no. 2Link zur Publikation , Link zum Preprint

### Harmonic Spinors for Twisted Dirac Operators

#### Autoren: Christian Bär (1997)

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, Spinc, or Spinh manifold are not topologically obstructed.

Zeitschrift:
Math. Ann.
Verlag:
Springer
Seiten:
225-246
Band:
309, no. 2

1996 | The Dirac Operator on Space Forms of Positive Curvature | Christian BärZeitschrift: J. Math. Soc. JapanVerlag: Mathematical Society of JapanSeiten: 69-83Band: 48, no. 1Link zur Publikation , Link zum Preprint

### The Dirac Operator on Space Forms of Positive Curvature

#### Autoren: Christian Bär (1996)

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.

Zeitschrift:
J. Math. Soc. Japan
Verlag:
Mathematical Society of Japan
Seiten:
69-83
Band:
48, no. 1

1996 | Harmonic Spinors and Topology | Christian BärVerlag: SpringerBuchtitel: New Developments in Differential GeometrySeiten: 53-66Link zur Publikation , Link zum Preprint

### Harmonic Spinors and Topology

#### Autoren: Christian Bär (1996)

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.

Verlag:
Springer
Buchtitel:
New Developments in Differential Geometry
Seiten:
53-66

1996 | Metrics with Harmonic Spinors | Christian BärZeitschrift: Geom. Funct. Anal.Verlag: SpringerSeiten: 899-942Band: 6, no. 6Link zur Publikation , Link zum Preprint

### Metrics with Harmonic Spinors

#### Autoren: Christian Bär (1996)

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.

Zeitschrift:
Geom. Funct. Anal.
Verlag:
Springer
Seiten:
899-942
Band:
6, no. 6

1994 | A Remark on Positively Curved 4-Manifolds | Christian BärZeitschrift: Differential Geom. Appl.Verlag: ElsevierSeiten: 71-75Band: 4, no. 1Link zur Publikation , Link zum Preprint

### A Remark on Positively Curved 4-Manifolds

#### Autoren: Christian Bär (1994)

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.

Zeitschrift:
Differential Geom. Appl.
Verlag:
Elsevier
Seiten:
71-75
Band:
4, no. 1

1993 | Elliptic Operators and Representation Theory of Compact Groups | Christian BärBuchtitel: Geometry and Global Analysis, Sendai 1993Seiten: 151-160Link zum Preprint

### Elliptic Operators and Representation Theory of Compact Groups

#### Autoren: Christian Bär (1993)

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.

Buchtitel:
Geometry and Global Analysis, Sendai 1993
Seiten:
151-160

1993 | Elliptische Operatoren und Darstellungstheorie kompakter Gruppen | Christian BärZeitschrift: Bonner Math. Schr.Reihe: Habilitationsschrift, Universität BonnSeiten: iv+50 pagesBand: 248Link zur Publikation

### Elliptische Operatoren und Darstellungstheorie kompakter Gruppen

#### Autoren: Christian Bär (1993)

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

Zeitschrift:
Bonner Math. Schr.
Reihe:
Habilitationsschrift, Universität Bonn
Seiten:
iv+50 pages
Band:
248

1993 | Real Killing Spinors and Holonomy | Christian BärZeitschrift: Comm. Math. Phys.Seiten: 509-521Band: 154, no. 3Link zur Publikation

### Real Killing Spinors and Holonomy

#### Autoren: Christian Bär (1993)

Zeitschrift:
Comm. Math. Phys.
Seiten:
509-521
Band:
154, no. 3

1992 | Harmonic Spinors on Riemann Surfaces | Christian Bär, Paul SchmutzZeitschrift: Ann. Global Anal. Geom.Verlag: SpringerSeiten: 263-273Band: 10, no. 3Link zur Publikation

### Harmonic Spinors on Riemann Surfaces

#### Autoren: Christian Bär, Paul Schmutz (1992)

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
263-273
Band:
10, no. 3

1992 | Lower Eigenvalue Estimates for Dirac Operators | Christian BärZeitschrift: Math. Ann.Verlag: SpringerSeiten: 39-46Band: 293, no. 1Link zur Publikation , Link zum Preprint

### Lower Eigenvalue Estimates for Dirac Operators

#### Autoren: Christian Bär (1992)

Zeitschrift:
Math. Ann.
Verlag:
Springer
Seiten:
39-46
Band:
293, no. 1

1992 | The Dirac Fundamental Tone of the Hyperbolic Space | Christian BärZeitschrift: Geom. DedicataVerlag: SpringerSeiten: 103-107Band: 41, no. 1Link zur Publikation

### The Dirac Fundamental Tone of the Hyperbolic Space

#### Autoren: Christian Bär (1992)

Zeitschrift:
Geom. Dedicata
Verlag:
Springer
Seiten:
103-107
Band:
41, no. 1

1992 | Upper Eigenvalue Estimates for Dirac Operators | Christian BärZeitschrift: Ann. Global Anal. Geom.Verlag: SpringerSeiten: 171-177Band: 10, no. 2Link zur Publikation

### Upper Eigenvalue Estimates for Dirac Operators

#### Autoren: Christian Bär (1992)

Zeitschrift:
Ann. Global Anal. Geom.
Verlag:
Springer
Seiten:
171-177
Band:
10, no. 2

1992 | The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces | Christian BärZeitschrift: Arch. Math.Verlag: SpringerSeiten: 65-79Band: 59, no. 1Link zur Publikation

### The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces

#### Autoren: Christian Bär (1992)

Zeitschrift:
Arch. Math.
Verlag:
Springer
Seiten:
65-79
Band:
59, no. 1

1992 | The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces (Erratum) | Christian BärLink zum Preprint

### The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces (Erratum)

#### Autoren: Christian Bär (1992)

1991 | Das Spektrum von Dirac-Operatoren | Christian BärZeitschrift: Bonner Math. Schr.Reihe: Dissertation, Universität BonnSeiten: xii+124 pagesBand: 217Link zum Preprint

### Das Spektrum von Dirac-Operatoren

#### Autoren: Christian Bär (1991)

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

Zeitschrift:
Bonner Math. Schr.
Reihe:
Dissertation, Universität Bonn
Seiten:
xii+124 pages
Band:
217