379 research outputs found

    Charge Conjugation from Space-Time Inversion

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    We show that the CPT group of the Dirac field emerges naturally from the PT and P (or T) subgroups of the Lorentz group.Comment: 4 pages, no figure

    Global Spinors and Orientable Five-Branes

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    Fermion fields on an M-theory five-brane carry a representation of the double cover of the structure group of the normal bundle. It is shown that, on an arbitrary oriented Lorentzian six-manifold, there is always an Sp(2) twist that allows such spinors to be defined globally. The vanishing of the arising potential obstructions does not depend on spin structure in the bulk, nor does the six-manifold need to be spin or spin-C. Lifting the tangent bundle to such a generalised spin bundle requires picking a generalised spin structure in terms of certain elements in the integral and modulo-two cohomology of the five-brane world-volume in degrees four and five, respectively.Comment: 18 pages, LaTeX; v2: version to appear in JHE

    Vafa-Witten Estimates for Compact Symmetric Spaces

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    We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.Comment: LaTeX, 11 pages. V2: Rigidity statement added, minor changes. To appea

    A characterization of quadric constant mean curvature hypersurfaces of spheres

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    Let ϕ:MSn+1Rn+2\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2} be an immersion of a complete nn-dimensional oriented manifold. For any vRn+2v\in\mathbb{R}^{n+2}, let us denote by v:MR\ell_v:M\to\mathbb{R} the function given by v(x)=ϕ(x),v\ell_v(x)=\phi(x),v and by fv:MRf_v:M\to\mathbb{R}, the function given by fv(x)=ν(x),vf_v(x)=\nu(x),v, where ν:MSn\nu:M\to\mathbb{S}^{n} is a Gauss map. We will prove that if MM has constant mean curvature, and, for some v0v\ne{\bf 0} and some real number λ\lambda, we have that v=λfv\ell_v=\lambda f_v, then, ϕ(M)\phi(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface MnM^n in Sn+1\mathbb{S}^{n+1} which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+42n+4.Comment: Final version (February 2008). To appear in the Journal of Geometric Analysi

    Generalised G2G_2-manifolds

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    We define new Riemannian structures on 7-manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2G_2, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by the means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in supergravity theory of type IIA/B with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skew torsion. Finally, we construct explicit examples by using the device of T-duality.Comment: 27 pages. v2: references added. v3: wrong argument (Theorem 3.3) and example (Section 4.1) removed, further examples added, notation simplified, all comments appreciated. v4:computation of Ricci tensor corrected, various minor changes, final version of the paper to appear in Comm. Math. Phy

    A volumetric Penrose inequality for conformally flat manifolds

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    We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to RnΩ,n3\R^{n}\setminus \Omega, n\ge 3, and so that their boundary is a minimal hypersurface. (Here, ΩRn\Omega\subset \R^{n} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by (V/βn)(n2)/n(V/\beta_{n})^{(n-2)/n}, where VV is the Euclidean volume of Ω\Omega and βn\beta_{n} is the volume of the Euclidean unit nn-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.Comment: 7 page

    Minimal cubic cones via Clifford algebras

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    We construct two infinite families of algebraic minimal cones in RnR^{n}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n4n\ge4, n16k+1n\ne 16k+1, there is at least one minimal cone in RnR^{n} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in Rm2R^{m^2}, m2m\ge2, defined by an irreducible homogeneous polynomial of degree mm. These examples provide particular answers to the questions on algebraic minimal cones posed by Wu-Yi Hsiang in the 1960's.Comment: Final version, corrects typos in Table

    Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds

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    We extend classical Euclidean stability theorems corresponding to the nonrelativistic Hamiltonians of ions with one electron to the setting of non parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar

    Rigidity of minimal surfaces in S 3

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    Isometric deformations of compact minimal surfaces in the standard three-sphere are studied. It is shown that a given surface admits only finitely many noncongruent minimal immersions into S 3 with the same first fundamental form.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46646/1/229_2005_Article_BF01258661.pd

    Generic metrics and the mass endomorphism on spin three-manifolds

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    Let (M,g)(M,g) be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point pMp\in M is called the mass endomorphism in pp associated to the metric gg due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.Comment: 8 page
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