125,774 research outputs found
The conformal transformation's controversy: what are we missing?
An alternative interpretation of the conformal transformations of the metric
is discussed according to which the latter can be viewed as a mapping among
Riemannian and Weyl-integrable spaces. A novel aspect of the conformal
transformation's issue is then revealed: these transformations relate
complementary geometrical pictures of a same physical reality, so that, the
question about which is the physical conformal frame, does not arise. In
addition, arguments are given which point out that, unless a clear statement of
what is understood by "equivalence of frames" is made, the issue is a semantic
one. For definiteness, an intuitively "natural" statement of conformal
equivalence is given, which is associated with conformal invariance of the
field equations. Under this particular reading, equivalence can take place only
if the metric is defined up to a conformal equivalence class. A concrete
example of a conformal-invariant theory of gravity is then explored. Since
Brans-Dicke theory is not conformally invariant, then the Jordan's and
Einstein's frames of the theory are not equivalent. Otherwise, in view of the
alternative approach proposed here, these frames represent complementary
geometrical descriptions of a same phenomenon. The different points of view
existing in the literature are critically scrutinized on the light of the new
arguments.Comment: 17 pages, no figures. version accepted by General Relativity and
Gravitation journa
Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichm\"uller Space
There exists on each Teichm\"uller space (comprising compact Riemann
surfaces of genus ), a natural sequence of determinant (of cohomology) line
bundles, , related to each other via certain ``Mumford isomorphisms''.
There is a remarkable connection, (Belavin-Knizhnik), between the Mumford
isomorphisms and the existence of the Polyakov string measure on the
Teichm\"uller space. This suggests the question of finding a genus-independent
formulation of these bundles and their isomorphisms. In this paper we combine a
Grothendieck-Riemann-Roch lemma with a new concept of bundles
to construct such an universal version. Our universal objects exist over the
universal space, , which is the direct limit of the as the
genus varies over the tower of all unbranched coverings of any base surface.
The bundles and the connecting isomorphisms are equivariant with respect to the
natural action of the universal commensurability modular group.Comment: ACTA MATHEMATICA (to appear); finalised version with a note of
clarification regarding the connection of the commensurability modular group
with the virtual automorphism group of the fundamental group of a closed
Riemann surface; 25 pages. LATE
Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary
Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is
quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with
Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and
isometrically on hyperbolic 3-space.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper7.abs.htm
The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems
We investigate wether three statements in analysis, that can be proved
classically, are realizable in the realizability model of extensional
continuous functionals induced by Kleene's second model . We prove that a
formulation of the Riemann Permutation Theorem as well as the statement that
all partially Cauchy sequences are Cauchy cannot be realized in this model,
while the statement that the product of two anti-Specker spaces is anti-Specker
can be realized
Kahler-Einstein metrics with edge singularities
This article considers the existence and regularity of Kahler-Einstein
metrics on a compact Kahler manifold with edge singularities with cone
angle along a smooth divisor . We prove existence of such
metrics with negative, zero and some positive cases for all cone angles
. The results in the positive case parallel those in the
smooth case. We also establish that solutions of this problem are
polyhomogeneous, i.e., have a complete asymptotic expansion with smooth
coefficients along for all .Comment: with an appendix by Chi Li and Yanir A. Rubinstein. Accepted by
Annals of Mat
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