1,501 research outputs found
Homogeneity of infinite dimensional anti-Kaehler isoparametric submanifolds II
In this paper, we prove that, if a full irreducible infinite dimensional
anti-Kaehler isoparametric submanifold of codimension greater than one has
-diagonalizable shape operators, then it is an orbit of the action of a
Banach Lie group generated by one-parameter transformation groups induced by
holomorphic Killing vector fields defined entirely on the ambient Hilbert
space.Comment: 40pages. arXiv admin note: substantial text overlap with
arXiv:1209.193
Renormalized Volume
We develop a universal distributional calculus for regulated volumes of
metrics that are singular along hypersurfaces. When the hypersurface is a
conformal infinity we give simple integrated distribution expressions for the
divergences and anomaly of the regulated volume functional valid for any choice
of regulator. For closed hypersurfaces or conformally compact geometries,
methods from a previously developed boundary calculus for conformally compact
manifolds can be applied to give explicit holographic formulae for the
divergences and anomaly expressed as hypersurface integrals over local
quantities (the method also extends to non-closed hypersurfaces). The resulting
anomaly does not depend on any particular choice of regulator, while the
regulator dependence of the divergences is precisely captured by these
formulae. Conformal hypersurface invariants can be studied by demanding that
the singular metric obey, smoothly and formally to a suitable order, a Yamabe
type problem with boundary data along the conformal infinity. We prove that the
volume anomaly for these singular Yamabe solutions is a conformally invariant
integral of a local Q-curvature that generalizes the Branson Q-curvature by
including data of the embedding. In each dimension this canonically defines a
higher dimensional generalization of the Willmore energy/rigid string action.
Recently Graham proved that the first variation of the volume anomaly recovers
the density obstructing smooth solutions to this singular Yamabe problem; we
give a new proof of this result employing our boundary calculus. Physical
applications of our results include studies of quantum corrections to
entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk
geometry, improved discussion of hypersurfaces with boundar
Almost Extrinsically Homogeneous Submanifolds of Euclidean Space
Consider a closed manifold M immersed in Rm. Suppose that the trivial bundle M × Rm = T M ⊗ ν M is equipped with an almost metric connection ~ ∇ which almost preserves the decomposition of M × Rm into the tangent and the normal bundle. Assume moreover that the difference Γ = ∂~∇ with the usual derivative ∂ in Rm is almost ~∇-parallel. Then M admits an extrinsically homogeneous immersion into Rm. Mathematics Subject Classifications (2000): 53C20, 53C24, 53C30, 53C42, 53C4
A positive Bondi--type mass in asymptotically de Sitter spacetimes
The general structure of the conformal boundary of
asymptotically de Sitter spacetimes is investigated. First we show that
Penrose's quasi-local mass, associated with a cut of the conformal
boundary, can be zero even in the presence of outgoing gravitational radiation.
On the other hand, following a Witten--type spinorial proof, we show that an
analogous expression based on the Nester--Witten form is finite only if the
Witten spinor field solves the 2-surface twistor equation on , and it
yields a positive functional on the 2-surface twistor space on ,
provided the matter fields satisfy the dominant energy condition. Moreover,
this functional is vanishing if and only if the domain of dependence of the
spacelike hypersurface which intersects in the cut
is locally isometric to the de Sitter spacetime. For non-contorted cuts this
functional yields an invariant analogous to the Bondi mass.Comment: 51 pages; typos corrected, one reference added; final version,
appearing in Class. Quantum Gra
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