379 research outputs found

### Charge Conjugation from Space-Time Inversion

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

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

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

Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a
complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let
us denote by $\ell_v:M\to\mathbb{R}$ the function given by
$\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by
$f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove
that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some
real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ 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 $M^n$ in $\mathbb{S}^{n+1}$
which is neither totally umbilical nor a Clifford hypersurface and has constant
scalar curvature is greater than or equal to $2n+4$.Comment: Final version (February 2008). To appear in the Journal of Geometric
Analysi

### Generalised $G_2$-manifolds

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 $G_2$, 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

We consider asymptotically flat Riemannian manifolds with nonnegative scalar
curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that
their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open
bounded with smooth mean-convex boundary.) We prove that the ADM mass of any
such manifold is bounded below by $(V/\beta_{n})^{(n-2)/n}$, where $V$ is the
Euclidean volume of $\Omega$ and $\beta_{n}$ is the volume of the Euclidean
unit $n$-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

We construct two infinite families of algebraic minimal cones in $R^{n}$. 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 $n\ge4$, $n\ne 16k+1$, there is
at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic
polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$,
defined by an irreducible homogeneous polynomial of degree $m$. 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

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

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

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the
expansion of the Green function for the Dirac operator at a fixed point $p\in
M$ is called the mass endomorphism in $p$ associated to the metric $g$ 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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