Prof. Dr. Christian Bär

Professor

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

Office hours

On Tuesdays 1-2 pm in room 0.18, house 9, Campus Golm

2024 | Local Index Theory for Lorentzian Manifolds | Christian Bär and Alexander StrohmaierZeitschrift: Annales scientifiques de l'École normale supérieureSeiten: 1693-1752Band: 57Link zur Publikation , Link zum Preprint

Local Index Theory for Lorentzian Manifolds

Autoren: Christian Bär and Alexander Strohmaier (2024)

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.

Zeitschrift:
Annales scientifiques de l'École normale supérieure
Seiten:
1693-1752
Band:
57

2024 | K-cowaist of manifolds with boundary | Christian Bär, Bernhard HankeZeitschrift: Comptes Rendus MathématiqueSeiten: 1349-1356Band: 362Link zur Publikation , Link zum Preprint

K-cowaist of manifolds with boundary

Autoren: Christian Bär, Bernhard Hanke (2024)

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

Zeitschrift:
Comptes Rendus Mathématique
Seiten:
1349-1356
Band:
362

2024 | Rigidity results for initial data sets satisfying the dominant energy condition | Christian Bär, Simon Brendle, Tsz-Kiu Aaron Chow, Bernhard HankeLink zum Preprint

Rigidity results for initial data sets satisfying the dominant energy condition

Autoren: Christian Bär, Simon Brendle, Tsz-Kiu Aaron Chow, Bernhard Hanke (2024)

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.

2024 | Dirac eigenvalues and the hyperspherical radius | Christian BärLink zum Preprint

Dirac eigenvalues and the hyperspherical radius

Autoren: Christian Bär (2024)

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.

2024 | Scalar curvature rigidity of warped product metrics | Christian Bär, Simon Brendle, Bernhard Hanke, Yipeng WangZeitschrift: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)Seiten: article 035, 26 pagesBand: 20Link zur Publikation , Link zum Preprint

Scalar curvature rigidity of warped product metrics

Autoren: Christian Bär, Simon Brendle, Bernhard Hanke, Yipeng Wang (2024)

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.

Zeitschrift:
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Seiten:
article 035, 26 pages
Band:
20

2024 | First-order elliptic boundary value problems on manifolds with non-compact boundary | Christian Bär und Lashi BandaraLink zum Preprint

First-order elliptic boundary value problems on manifolds with non-compact boundary

Autoren: Christian Bär und Lashi Bandara (2024)

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.

2023 | Boundary conditions for scalar curvature | Christian Bär and Bernhard HankeVerlag: World ScientificBuchtitel: M. Gromov, B. Lawson (eds): Perspectives in Scalar CurvatureSeiten: 325-377Band: 2Link zur Publikation , Link zum Preprint

Boundary conditions for scalar curvature

Autoren: Christian Bär and Bernhard Hanke (2023)

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.

Verlag:
World Scientific
Buchtitel:
M. Gromov, B. Lawson (eds): Perspectives in Scalar Curvature
Seiten:
325-377
Band:
2

2022 | The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions | Christian Bär and Penelope GehringLink zum Preprint

The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions

Autoren: Christian Bär and Penelope Gehring (2022)

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.
 

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

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