176 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
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
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
A characterization of quadric constant mean curvature hypersurfaces of spheres
Let be an immersion of a
complete -dimensional oriented manifold. For any , let
us denote by the function given by
and by , the function given by
, where is a Gauss map. We will prove
that if has constant mean curvature, and, for some and some
real number , we have that , then, 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 in
which is neither totally umbilical nor a Clifford hypersurface and has constant
scalar curvature is greater than or equal to .Comment: Final version (February 2008). To appear in the Journal of Geometric
Analysi
Generalised -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 , 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 , and so that
their boundary is a minimal hypersurface. (Here, is open
bounded with smooth mean-convex boundary.) We prove that the ADM mass of any
such manifold is bounded below by , where is the
Euclidean volume of and is the volume of the Euclidean
unit -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
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
Minimal immersions of closed surfaces in hyperbolic three-manifolds
We study minimal immersions of closed surfaces (of genus ) in
hyperbolic 3-manifolds, with prescribed data , where
is a conformal structure on a topological surface , and is a holomorphic quadratic differential on the surface . We
show that, for each for some , depending only on
, there are at least two minimal immersions of closed surface
of prescribed second fundamental form in the conformal structure
. Moreover, for sufficiently large, there exists no such minimal
immersion. Asymptotically, as , the principal curvatures of one
minimal immersion tend to zero, while the intrinsic curvatures of the other
blow up in magnitude.Comment: 16 page
Willmore Surfaces of Constant Moebius Curvature
We study Willmore surfaces of constant Moebius curvature in . It is
proved that such a surface in must be part of a minimal surface in
or the Clifford torus. Another result in this paper is that an isotropic
surface (hence also Willmore) in of constant could only be part of a
complex curve in or the Veronese 2-sphere in . It is
conjectured that they are the only examples possible. The main ingredients of
the proofs are over-determined systems and isoparametric functions.Comment: 16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6)
has been correcte
-minimal surface and manifold with positive -Bakry-\'{E}mery Ricci curvature
In this paper, we first prove a compactness theorem for the space of closed
embedded -minimal surfaces of fixed topology in a closed three-manifold with
positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type
lower bound of the first eigenvalue of the -Laplacian on compact manifold
with positive -Bakry-\'{E}mery Ricci curvature, and prove that the lower
bound is achieved only if the manifold is isometric to the -shpere, or the
-dimensional hemisphere. Finally, for compact manifold with positive
-Bakry-\'{E}mery Ricci curvature and -mean convex boundary, we prove an
upper bound for the distance function to the boundary, and the upper bound is
achieved if only if the manifold is isometric to an Euclidean ball.Comment: 15 page
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