@article{BaerBandara2022, author = {B{\"a}r, Christian and Bandara, Lashi}, title = {Boundary value problems for general first-order elliptic differential operators}, series = {Journal of functional analysis}, volume = {282}, journal = {Journal of functional analysis}, number = {12}, publisher = {Elsevier}, address = {Amsterdam [u.a.]}, issn = {0022-1236}, doi = {10.1016/j.jfa.2022.109445}, pages = {69}, year = {2022}, abstract = {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 show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.}, language = {en} } @article{BaerHanke2022, author = {B{\"a}r, Christian and Hanke, Bernhard}, title = {Local flexibility for open partial differential relations}, series = {Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ.}, volume = {75}, journal = {Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ.}, number = {6}, publisher = {Wiley}, address = {Hoboken}, issn = {0010-3640}, doi = {10.1002/cpa.21982}, pages = {1377 -- 1415}, year = {2022}, abstract = {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 that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general.}, language = {en} } @article{BaerMazzeo2021, author = {B{\"a}r, Christian and Mazzeo, Rafe}, title = {Manifolds with many Rarita-Schwinger fields}, series = {Communications in mathematical physics}, volume = {384}, journal = {Communications in mathematical physics}, number = {1}, publisher = {Springer}, address = {Berlin}, issn = {0010-3616}, doi = {10.1007/s00220-021-04030-0}, pages = {533 -- 548}, year = {2021}, abstract = {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 Kahler-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.}, language = {en} } @article{Baer2021, author = {B{\"a}r, Christian}, title = {The Faddeev-LeVerrier algorithm and the Pfaffian}, series = {Linear algebra and its applications}, volume = {630}, journal = {Linear algebra and its applications}, publisher = {Elsevier}, address = {New York}, issn = {0024-3795}, doi = {10.1016/j.laa.2021.07.023}, pages = {39 -- 55}, year = {2021}, abstract = {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(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). 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 using computer algebra.}, language = {en} } @article{Baer2019, author = {B{\"a}r, Christian}, title = {The curl operator on odd-dimensional manifolds}, series = {Journal of mathematical physics}, volume = {60}, journal = {Journal of mathematical physics}, number = {3}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/1.5082528}, pages = {16}, year = {2019}, abstract = {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. Published under license by AIP Publishing.}, language = {en} } @article{BaerStrohmaier2019, author = {B{\"a}r, Christian and Strohmaier, Alexander}, title = {An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary}, series = {American Journal of Mathematics}, volume = {141}, journal = {American Journal of Mathematics}, number = {5}, publisher = {Johns Hopkins Univ. Press}, address = {Baltimore}, issn = {0002-9327}, doi = {10.1353/ajm.2019.0037}, pages = {1421 -- 1455}, year = {2019}, abstract = {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.}, language = {en} } @article{BaerStrohmaier2016, author = {B{\"a}r, Christian and Strohmaier, Alexander}, title = {A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds}, series = {Communications in mathematical physics}, volume = {347}, journal = {Communications in mathematical physics}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-016-2664-1}, pages = {703 -- 721}, year = {2016}, abstract = {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 directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the.-invariant of the Cauchy hypersurfaces.}, language = {en} } @article{BaerWafo2015, author = {B{\"a}r, Christian and Wafo, Roger Tagne}, title = {Initial value problems for wave equations on manifolds}, series = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, volume = {18}, journal = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {1385-0172}, doi = {10.1007/s11040-015-9176-7}, pages = {29}, year = {2015}, abstract = {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 Hormander.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes}, series = {Communications in mathematical physics}, volume = {333}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-014-2097-7}, pages = {1585 -- 1615}, year = {2015}, abstract = {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.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Geometrically formal 4-manifolds with nonnegative sectional curvature}, series = {Communications in analysis and geometry}, volume = {23}, journal = {Communications in analysis and geometry}, number = {3}, publisher = {International Press of Boston}, address = {Somerville}, issn = {1019-8385}, pages = {479 -- 497}, year = {2015}, abstract = {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 CP2. 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.}, language = {en} } @article{BaerBecker2014, author = {B{\"a}r, Christian and Becker, Christian}, title = {Differential characters and geometric chains}, series = {Lecture notes in mathematics : a collection of informal reports and seminars}, volume = {2112}, journal = {Lecture notes in mathematics : a collection of informal reports and seminars}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-319-07034-6; 978-3-319-07033-9}, issn = {0075-8434}, doi = {10.1007/978-3-319-07034-6_1}, pages = {1 -- 90}, year = {2014}, abstract = {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.}, language = {en} } @article{BaerPfaeffle2010, author = {B{\"a}r, Christian and Pfaeffle, Frank}, title = {Asymptotic heat kernel expansion in the semi-classical limit}, issn = {0010-3616}, doi = {10.1007/s00220-009-0973-3}, year = {2010}, abstract = {Let H-h = h(2)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 SE arrow 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.}, language = {en} } @article{BaerBessa2010, author = {B{\"a}r, Christian and Bessa, C. Pacelli}, title = {Stochastic completeness and volume growth}, issn = {0002-9939}, doi = {10.1090/S0002-9939-10-10281-0}, year = {2010}, abstract = {It was 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 counterexamples to a converse implication.}, language = {en} } @article{Baer2009, author = {B{\"a}r, Christian}, title = {Spectral bounds for Dirac operators on open manifolds}, issn = {0232-704X}, doi = {10.1007/s10455-008-9149-1}, year = {2009}, abstract = {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.}, language = {en} } @article{BaerSchopka2003, author = {B{\"a}r, Christian and Schopka, Sven}, title = {The dirac determinant of spherical space forms}, year = {2003}, abstract = {The zeta-regularized determinants of the Dirac operator and of its square are computed on spherical space forms. On S^2 the determinant of Dirac operators twisted by a complex line bundle is also calculated.}, language = {en} } @article{BaerDahl2003, author = {B{\"a}r, Christian and Dahl, Matthias}, title = {Small eigenvalues of the conformal laplacian}, year = {2003}, abstract = {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 alpha-genus.}, language = {en} } @article{BaerMoroianu2003, author = {B{\"a}r, Christian and Moroianu, Sergiu}, title = {Heat kernel asymptotics for roots of generalized laplacians}, year = {2003}, abstract = {We describe the heat kernel asymptotics for roots of a Laplace type operator on a closed manifold. A previously known relation between the Wodzicki residue and heat trace asymptotics is shown to hold pointwise for the corresponding densities.}, language = {en} } @article{BaerDahl2004, author = {B{\"a}r, Christian and Dahl, Matthias}, title = {The first dirac eigenvalue on manifolds with positive scalar curvature}, year = {2004}, abstract = {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.}, language = {en} } @article{BaerGouduchonMoroianu2005, author = {B{\"a}r, Christian and Gouduchon, Paul and Moroianu, Andrei}, title = {Generalized Cylinders in Semi-Riemannian and Spin Geometry}, year = {2005}, abstract = {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.}, language = {en} } @book{BaerGinouxPfaeffle2007, author = {B{\"a}r, Christian and Ginoux, Nicolas and Pf{\"a}ffle, Frank}, title = {Wave equations on lorentzian manifolds and quantization}, publisher = {European Math. Society}, address = {Z{\"u}rich}, isbn = {978-3-03719-037-1}, pages = {194 S.}, year = {2007}, language = {en} }