Search

Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4. [ABSTRACT FROM AUTHOR]

On metrics of Eguchi-Hanson type II with negative constant Ricci curvatures, the authors show that there is no nontrivial Killing spinor. On metrics of Eguchi-Hanson type II with negative constant scalar curvature, they show that there is no nontrivial Lp eigenspinor for 0 < p < 2 if the eigenvalue has nontrivial real part, and no nontrivial L2 eigenspinor if either the eigenvalue has trivial real part or the eigenvalue is real, the eigenspinor is isotropic and the parameter η in radial and angular equations for eigenspinors is real. They also solve harmonic spinors and eigenspinors explicitly on metrics of Eguchi-Hanson type II with certain special potentials. [ABSTRACT FROM AUTHOR]

We obtain optimal lower bounds for the eigenvalues of the Dirac–Witten operator on locally reducible spacelike submanifolds in terms of intrinsic and extrinsic quantities. The limiting cases are also studied. [ABSTRACT FROM AUTHOR]

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface Σ bounding a noncompact domain in a spin asymptotically flat manifold (M n , g) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of Σ in (M n , g) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest. [ABSTRACT FROM AUTHOR]

We examine which of the compact connected Lie groups that act transitively on spheres of different dimensions leave the unique spin structure of the sphere invariant. We study the notion of invariance of a spin structure and prove this classification in two different ways; through examining the differential of the actions and through representation theory. [ABSTRACT FROM AUTHOR]

We study the influence of an additional scalar potential on various geometric and analytic properties of Dirac-harmonic maps. We will create a mathematical wish list of the possible benefits from inducing the potential term and point out that the latter cannot be achieved in general. Finally, we focus on several potentials that are motivated from supersymmetric quantum field theory. [ABSTRACT FROM AUTHOR]

The Riemannian product M 1 (c 1) × M 2 (c 2) , where M i (c i) denotes the 2-dimensional space form of constant sectional curvature c i ∈ R , has two different Spin c structures carrying each a parallel spinor. The restriction of these two parallel spinor fields to a 3-dimensional hypersurface M characterizes the isometric immersion of M into M 1 (c 1) × M 2 (c 2) . As an application, we prove that totally umbilical hypersurfaces of M 1 (c 1) × M 1 (c 1) and totally umbilical hypersurfaces of M 1 (c 1) × M 2 (c 2) ( c 1 ≠ c 2 ) having a local structure product are of constant mean curvature. [ABSTRACT FROM AUTHOR]

In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin j + 1 / 2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power j + 1 and understand how the spinor fields with spin j + 1 / 2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor ψ is a spinor that satisfies the equation ∇ X ψ = A X · ψ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R × N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to S 2 × R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe. [ABSTRACT FROM AUTHOR]

Let (M i , g i) i ∈ N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov–Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D B on B. We give an explicit description of D B and characterize the special case where D B equals the Dirac operator on B. [ABSTRACT FROM AUTHOR]

We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the ν -entropy of Perelman and hence is dynamically unstable for the Ricci flow. [ABSTRACT FROM AUTHOR]

We discuss a method to construct Dirac-harmonic maps developed by Jost et al. (J Geom Phys 59(11):1512–1527, 2009). The method uses harmonic spinors and twistor spinors and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in Jost et al. (2009) were fulfilled. We show that for isometric immersions into space forms, these conditions are fulfilled only under special assumptions. In several cases, we show the existence of solutions. [ABSTRACT FROM AUTHOR]

We introduce in this paper an adjustment for the Webster-Tanaka connection of pseudo-Hermitian geometry on CR manifolds, which behaves naturally with respect to issues concerning parallel sections and holonomy. We call this connection and its holonomy group basic. The basic holonomy group is suitable to describe pseudo-Einstein spaces. Moreover, for pseudo-Hermitian spin manifolds, we discuss the existence of (transversally) parallel spinors. Such spinors exist on pseudo-Einstein spaces, no matter of the sign of the Webster scalar curvature. [ABSTRACT FROM AUTHOR]

In this article, we develop energy methods for a large class of linear and nonlinear Dirac-type equations in two-dimensional Minkowski space. We will derive existence results for several Dirac-type equations originating in quantum field theory, in particular for Dirac-wave maps to compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes. [ABSTRACT FROM AUTHOR]

We factorize the Dirac operator on the Connes–Landi 4-sphere in unbounded KK-theory. We show that a family of Dirac operators along the orbits of the torus action defines an unbounded Kasparov module, while the Dirac operator on the principal orbit space –an open quadrant in the 2-sphere – defines a half-closed chain. We show that the tensor sum of these two operators coincides up to unitary equivalence with the Dirac operator on the Connes–Landi sphere and prove that this tensor sum is an unbounded representative of the internal Kasparov product in bivariant K-theory. We also generalize our results to Dirac operators on all toric noncommutative manifolds subject to a condition on the principal stratum. We find that there is a curvature term that arises as an obstruction for having a tensor sum decomposition in unbounded KK-theory. This curvature term can however not be detected at the level of bounded KK-theory. [ABSTRACT FROM AUTHOR]

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spin c Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results. [ABSTRACT FROM AUTHOR]

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds. [ABSTRACT FROM AUTHOR]

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $$2n \ge 6$$ , the natural KO-orientation from the infinite loop space of the Madsen-Tillmann-Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2 n-dimensional spin manifold. For manifolds of odd dimension $$2n+1 \ge 7$$ , we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov-Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah-Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen-Weiss theorem, proven by Galatius and the third named author. [ABSTRACT FROM AUTHOR]

We show that the limit at infinity of the vector-valued Brown–York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic 3-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type formula expressing the asymptotic limit of this quasi-local mass as a bulk integral which manifestly has the right sign under the above assumptions. In the spirit of recent work by Hijazi, Montiel and Raulot, we also provide another proof of this result which uses the theory of boundary value problems for Dirac operators on compact domains to show that a certain quasi-local mass, which converges to the Brown–York mass in the asymptotic limit, has the expected causal character under suitable geometric assumptions. [ABSTRACT FROM AUTHOR]

We study orientability issues of moduli spaces from gauge theories on Calabi–Yau manifolds. Our results generalize and strengthen those for Donaldson–Thomas theory on Calabi–Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing formula in DT theory. [ABSTRACT FROM AUTHOR]

We show that Killing tensors on conformally flat n -dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta. [ABSTRACT FROM AUTHOR]

The complex projective space C P 2 of complex dimension 2 has a Spin c structure carrying Kählerian Killing spinors. The restriction of one of these Kählerian Killing spinors to a surface M 2 characterizes the isometric immersion of M 2 into C P 2 if the immersion is either Lagrangian or complex. [ABSTRACT FROM AUTHOR]

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory. [ABSTRACT FROM AUTHOR]

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in R n and in the 3-dimensional Lie groups S 3 and E ( κ , τ ) , and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional metric Lie groups recently given by Meeks, Mira, Perez and Ros. [ABSTRACT FROM AUTHOR]

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow. [ABSTRACT FROM AUTHOR]

On Riemannian or spin manifolds, there are geometric first order differential operators called generalized gradients. In this article, we prove that the Dirac operator twisted with an associated bundle is a linear combination of some generalized gradients. This observation allows us to find all the homomorphism type Weitzenböck formulas. We also give some applications. [ABSTRACT FROM AUTHOR]

We show that any 4-dimensional Lorentzian manifold with nondegenerate Weyl tensor possesses the spinor structure and this structure is natural, i.e. invariant under action of diffeomorphisms. [ABSTRACT FROM AUTHOR]

We define (higher rank) spinorially twisted spin structures and deduce various curvature identities as well as estimates for the eigenvalues of the corresponding twisted Dirac operators. [ABSTRACT FROM AUTHOR]

The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained. [ABSTRACT FROM AUTHOR]

We study the evolution equations for a regularized version of Dirac-geodesics, which are the one-dimensional analogue of Dirac-harmonic maps. We show that for the regularization being sufficiently large, the evolution equations subconverge to a regularized Dirac-geodesic. Finally, we discuss the relation between the regularized and the original problem. [ABSTRACT FROM AUTHOR]

Dirac-geodesics are Dirac-harmonic maps from one dimensional domains. In this paper, we introduce the heat flow for Dirac-geodesics and establish its long-time existence and an asymptotic property of the global solution. We classify Dirac-geodesics on the standard 2-sphere $$S^2(1)$$ and the hyperbolic plane $$\mathbb {H}^2$$ , and derive existence results on topological spheres and hyperbolic surfaces. These solutions constitute new examples of coupled Dirac-harmonic maps (in the sense that the map part is not simply a harmonic map). [ABSTRACT FROM AUTHOR]

We prove that an isometric immersion of a simply connected Lorentzian surface in R 2 , 2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in R 2 , 2 to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in R 2 , 2 , and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant. [ABSTRACT FROM AUTHOR]

We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally isotropic subspace of arbitrary dimension, there is, w.r.t. a local metric in the conformal class defined off a singular set, a parallel, totally isotropic distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant isotropic lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification of geometries admitting twistor spinors described in terms of parallel spin tractors using conformal spin tractor calculus. As an example we obtain together with already known results about generic distributions in dimensions 5 and 6 a complete geometric description of local geometries admitting real twistor spinors in signatures ( 3 , 2 ) and ( 3 , 3 ) . In contrast to the generic case where generic geometric distributions play an important role, the underlying geometries in the non-generic case without zeroes turn out to admit integrable distributions. [ABSTRACT FROM AUTHOR]

We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces. [ABSTRACT FROM AUTHOR]

Suppose that Σ = ∂ Ω is the n -dimensional boundary, with positive (inward) mean curvature H , of a connected compact ( n + 1 ) -dimensional Riemannian spin manifold ( Ω n + 1 , g ) whose scalar curvature R ≥ − n ( n + 1 ) k 2 , for some k > 0 . If Σ admits an isometric and isospin immersion F into the hyperbolic space H − k 2 n + 1 , we define a quasi-local mass and prove its positivity as well as the associated rigidity statement. The proof is based on a holographic principle for the existence of an imaginary Killing spinor. For n = 2 , we also show that its limit, for coordinate spheres in an Asymptotically Hyperbolic (AH) manifold, is the mass of the (AH) manifold. [ABSTRACT FROM AUTHOR]

On a closed, connected Riemannian manifold with a Kähler foliation of codimension q = 2 m , any transverse Killing r ( ≥ 2 ) -form is parallel (Jung and Jung, 2012). In this paper, we study transverse conformal Killing forms on Kähler foliations. In fact, if the foliation is minimal, then for any transverse conformal Killing r -form ϕ ( 2 ≤ r ≤ q − 2 ) , J ϕ is parallel. Here J is defined in Section 4 . [ABSTRACT FROM AUTHOR]

We prove that, among all $$(n+1)$$ -dimensional spin static vacua with positive cosmological constant, the de Sitter spacetime is characterized by the fact that its spatial Killing horizons have minimal modes for the Dirac operator. As a consequence, the de Sitter spacetime is the only vacuum of this type for which the induced metric tensor on some of its Killing horizons is at least equal to that of a round $$(n-1)$$ -sphere. This extends uniqueness theorems shown in Boucher et al. (Phys Rev D 30:2447, ). Chruściel []. to more general horizon metrics and to the non-single horizon case. [ABSTRACT FROM AUTHOR]

We construct a two parameter family of eleven-dimensional indecomposable Cahen–Wallach spaces with irreducible, non-flat, non-restricted geometric supersymmetry of fraction ν = 3 4 . Its compactified moduli space can be parameterized by a compact interval with two points corresponding to two non-isometric, decomposable spaces. These singular spaces are associated to a restricted N = 4 geometric supersymmetry with ν = 1 2 in dimension six and a non-restricted N = 2 geometric supersymmetry with ν = 3 4 in dimension nine. [ABSTRACT FROM AUTHOR]

We study generalized Killing spinors on the standard sphere S 3 , which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold S 3 × S 3 and to great circle flows on S 3 . Using our methods we generalize a well known result of Gluck and Gu [5] concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of S 3 × S 3 has at least three connected components. [ABSTRACT FROM AUTHOR]

We establish a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition and then give various Omori-Yau type maximum principles on complete noncompact manifolds. We also obtain Liouville theorems for V-harmonic maps. We apply these findings to Ricci solitons and self-shrinkers. [ABSTRACT FROM AUTHOR]

We study generalized Killing spinors on round spheres $$\mathbb {S}^n$$ . We show that on the standard sphere $$\mathbb {S}^8$$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on $$\mathbb {S}^n$$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola-Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors. [ABSTRACT FROM AUTHOR]

We consider a two-parameter generalization $$D_{ab}$$ of the Riemann Dirac operator $$D$$ on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for $$D_{ab}$$ . We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of $$D_{ab}$$ , we prove several eigenvalue estimates for $$D_{ab}$$ which improve Friedrich's estimate (Friedrich, Math Nachr 97, 117-146, ). [ABSTRACT FROM AUTHOR]

It is well known that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article, we extend this result to a global version. We view the spectrum of a Dirac operator as a function $$\mathbb Z \,\rightarrow \mathbb R \,$$ Z → R and endow the space of all spectra with an $$\mathrm{arsinh }$$ arsinh -uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that, due to spectral flow, these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms in general. [ABSTRACT FROM AUTHOR]

In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterizations of surfaces in $$\mathbb R ^3$$ R 3 and in $$S^3$$ S 3 given by Friedrich and by Morel. [ABSTRACT FROM AUTHOR]

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions. [ABSTRACT FROM AUTHOR]

The positive energy theorem is a significant subject in general relativity theory. In Witten’s proof of this theorem, the solution of a free Dirac equation which is a spinor filed plays an important role. In order to prove the positive energy theorem for black holes, Gibbons, Hawking, Horowitz and Perry imposed a local boundary condition on the apparent horizon of the black hole. Then the Dirac equation under this boundary condition forms an elliptic boundary value problem. In fact, this kind of local boundary condition can be generally defined by a Chirality operator on the Dirac bundle over a spin manifold. In this paper, by establishing a proper analysis setting and developing variational arguments, we study a nonlinear Dirac equation on a compact spin manifold (M, g) which satisfies the local boundary condition with respect to a Chirality operator. [ABSTRACT FROM AUTHOR]