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Dirac-geodesics with curvature term are Dirac-harmonic maps with curvature term from one-dimensional domains. In this paper, we first describe the structure of Dirac-geodesics with curvature term in surfaces and give solutions on the unit 2-sphere and the hyperbolic plane, and then we give the structure of solutions in warped product spaces. Finally, we define the heat flow of Dirac-geodesics with curvature term and prove the global existence and sub-convergence of the heat flow into any closed surfaces and space forms. Our results provide the first examples and general existence theorems of Dirac-geodesics with curvature term. [ABSTRACT FROM AUTHOR]

In this paper, we prove the long neck principle, band width estimates, and width inequalities of the geodesic collar neighborhoods of the boundary in the setting of general initial data sets for the Einstein equations, subject to certain energy conditions corresponding to the lower bounds of scalar curvature on Riemannian manifolds. Our results are established via the spinorial Callias operator approach. [ABSTRACT FROM AUTHOR]

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]

We develop an invariant approach to SU(2)–structures on spin 5–manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to SU(2). Moreover, we show how to induce quaternionic structure on the contact distribution of the considered SU(2)–structure. We show the invariance of certain components of the covariant derivative ∇ φ , where φ is any spinor field defining SU(2)–structure. This shows, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. We conclude with the explicit description of the intrinsic torsion and the characteristic connection. [ABSTRACT FROM AUTHOR]

We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations, directly leading to the spinor representation of conformal surfaces in isotropic space. As an application, we obtain the Weierstrass-type representation for zero mean curvature surfaces, and the Kenmotsu-type representation for constant mean curvature surfaces, allowing us to construct many explicit examples. [ABSTRACT FROM AUTHOR]

The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its classical counterpart for Riemannian metrics. Recent non-triviality results for the latter will then imply that this space has non-trivial homotopy groups. [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 present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in S 2 × R and we obtain new spinorial characterizations of immersions in S 2 × R 2 and in H 2 × R. We then study the theory of H = 1 / 2 surfaces in H 2 × R using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of H = 1 / 2 surfaces in R 1 , 2 . [ABSTRACT FROM AUTHOR]

We define a relative L 2 - ρ -invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a 4 k + 3 -dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised η -invariants for odd-dimensional spin manifolds with boundary. [ABSTRACT FROM AUTHOR]

We define a variant of the Seiberg–Witten equations using the Rarita–Schwinger operators for a closed simply connected spin smooth 4-manifold X. The moduli space of solutions to the system of non-linear differential equations consists of harmonic 3/2-spinors and U(1)-connections satisfying certain curvature condition. Besides having an obvious U(1)-symmetry, these equations also have a symmetry by Pin(2). We exploit this additional symmetry to perform finite-dimensional approximations for the eigenvalue problem of the 3/2-monopole map and show that under a certain topological assumption, the moduli space of solutions is always non-compact and thus non-empty. [ABSTRACT FROM AUTHOR]

We consider the Positive Mass Theorem for Riemannian manifolds (M n , g) with the asymptotic end (R k × X n - k , g R k + g X) ( k ≥ 3 ) by studying the corresponding compactification problem. Here (X , g X) is a compact scalar flat manifold. We show that the Positive Mass Theorem holds if certain generalized connected sum admits no metric of positive scalar curvature. Moreover we establish the rigidity result, namely, the mass is zero iff M is isometric to R k × X n - k . [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]

Our main goal in the present paper is to expand the known class of noncompact manifolds over which the L 2 -spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the L p -spectrum of the Dirac operator and its square is independent of p for p ≥ 1 . Using the L 1 -spectrum, which is simpler to compute, we generalize the class of manifolds over which the L p -spectrum of the Dirac operator is the real line for all p. We also show that by applying the generalized Weyl criterion, we can find large classes of manifolds with asymptotically nonnegative Ricci curvature, or which are asymptotically flat, such that the L 2 -spectrum of a general Dirac operator and its square is maximal. [ABSTRACT FROM AUTHOR]

We consider a kind of generalized harmonic maps, namely, the VT-harmonic maps. We prove an existence theorem for the Dirichlet problem of VT-harmonic maps from compact manifolds with boundary. [ABSTRACT FROM AUTHOR]

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of S 7 -bundles over S 8 which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional curvature on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components. [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]

Let Z be a spin 4-manifold carrying a parallel spinor and M ↪ Z a hypersurface. The second fundamental form of the embedding induces a flat metric connection on TM. Such flat connections satisfy a non-elliptic, non-linear equation in terms of a symmetric 2-tensor on M. When M is compact and has positive scalar curvature, the linearized equation has finite-dimensional kernel. Four families of solutions are known on the round 3-sphere S 3 . We study the linearized equation in the vicinity of these solutions and we construct as a byproduct an incomplete hyperkähler metric on S 3 × R closely related to the Euclidean Taub-NUT metric on R 4 . On S 3 there do not exist other solutions which either are constant in a left (or right) invariant frame, have three distinct constant eigenvalues, or are invariant in the direction of a left (or right)-invariant eigenvector. We deduce from this last result an extension of Liebmann’s sphere rigidity theorem. [ABSTRACT FROM AUTHOR]

In this article, we study the stability problem for the Einstein–Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable. [ABSTRACT FROM AUTHOR]

We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit representations of the Lie algebras 𝔰𝔭𝔦𝔶 (8), 𝔰𝔭𝔦𝔶 (7) and 𝔤2, etc.

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]

We generalized the lower bound estimates for eigenvalues of the Dirac operator on spacelike hypersurfaces of Lorentzian manifolds obtained by Yongfa Chen in (Sci China Ser A Math 52(11):2459–2468, 2009) based on the constraint between the scalar curvature of the manifold, energy–momentum tensor and the mean curvature of the manifold. Afterwards, we examined the geometric data in the case of estimation satisfies equality condition. [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]