546 research outputs found
Mean Curvature Flow on Ricci Solitons
We study monotonic quantities in the context of combined geometric flows. In
particular, focusing on Ricci solitons as the ambient space, we consider
solutions of the heat type equation integrated over embedded submanifolds
evolving by mean curvature flow and we study their monotonicity properties.
This is part of an ongoing project with Magni and Mantegazzawhich will treat
the case of non-solitonic backgrounds \cite{a_14}.Comment: 19 page
Irreversibility of World-sheet Renormalization Group Flow
We demonstrate the irreversibility of a wide class of world-sheet
renormalization group (RG) flows to first order in in string theory.
Our techniques draw on the mathematics of Ricci flows, adapted to
asymptotically flat target manifolds. In the case of somewhere-negative scalar
curvature (of the target space), we give a proof by constructing an entropy
that increases monotonically along the flow, based on Perelman's Ricci flow
entropy. One consequence is the absence of periodic solutions, and we are able
to give a second, direct proof of this. If the scalar curvature is everywhere
positive, we instead construct a regularized volume to provide an entropy for
the flow. Our results are, in a sense, the analogue of Zamolodchikov's
-theorem for world-sheet RG flows on noncompact spacetimes (though our
entropy is not the Zamolodchikov -function).Comment: Minor changes, added one citation, version accepted for publicatio
A simple proof of Perelman's collapsing theorem for 3-manifolds
We will simplify earlier proofs of Perelman's collapsing theorem for
3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we
use Perelman's critical point theory (e.g., multiple conic singularity theory
and his fibration theory) for Alexandrov spaces to construct the desired local
Seifert fibration structure on collapsed 3-manifolds. The verification of
Perelman's collapsing theorem is the last step of Perelman's proof of
Thurston's Geometrization Conjecture on the classification of 3-manifolds. Our
proof of Perelman's collapsing theorem is almost self-contained, accessible to
non-experts and advanced graduate students. Perelman's collapsing theorem for
3-manifolds can be viewed as an extension of implicit function theoremComment: v1: 9 Figures. In this version, we improve the exposition of our
arguments in the earlier arXiv version. v2: added one more grap
Investigating Off-shell Stability of Anti-de Sitter Space in String Theory
We propose an investigation of stability of vacua in string theory by
studying their stability with respect to a (suitable) world-sheet
renormalization group (RG) flow. We prove geometric stability of (Euclidean)
anti-de Sitter (AdS) space (i.e., ) with respect to the simplest
RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point
of Ricci flow. We therefore choose an appropriate flow for which it is a fixed
point, prove a linear stability result for AdS space with respect to this flow,
and then show this implies its geometric stability with respect to Ricci flow.
The techniques used can be generalized to RG flows involving other fields. We
also discuss tools from the mathematics of geometric flows that can be used to
study stability of string vacua.Comment: 29 pages, references added in this version to appear in Classical and
Quantum Gravit
Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
We apply some methods of homology and K-theory to special classes of branes
wrapping homologically nontrivial cycles. We treat the classification of
four-geometries in terms of compact stabilizers (by analogy with Thurston's
classification of three-geometries) and derive the K-amenability of Lie groups
associated with locally symmetric spaces listed in this case. More complicated
examples of T-duality and topology change from fluxes are also considered. We
analyse D-branes and fluxes in type II string theory on with torsion flux and demonstrate in details
the conjectured T-duality to with no flux. In the
simple case of , T-dualizing the circles reduces to
duality between with
flux and with no flux.Comment: 27 pages, tex file, no figure
An evolution equation as the WKB correction in long-time asymptotics of Schrodinger dynamics
We consider 3d Schrodinger operator with long-range potential that has
short-range radial derivative. The long-time asymptotics of non-stationary
problem is studied and existence of modified wave operators is proved. It turns
out, the standard WKB correction should be replaced by the solution to certain
evolution equation.Comment: This is a preprint of an article whose final and definitive form has
been published in Comm. Partial Differential Equations, available online at
http://www.informaworld.co
a cross-sectional study in six European cities
Background: The relationship between socioeconomic position (SEP) and adolescent physical activity is uncertain, as most evidence is limited to specific settings and a restricted number of SEP indicators. This study aimed to assess the magnitude of socioeconomic differences in adolescent vigorous physical activity (VPA) across various European countries using a wide range of SEP indicators, including family-based (education, family affluence, perceived social standing, parents’ employment, housing tenure) and adolescent-based (academic performance and pocket money) ones. Methods: We used data from a survey among 10,510 students aged 14–17 from 50 schools in six European cities: Namur (BE), Tampere (FI), Hannover (DE), Latina (IT), Amersfoort (NL), Coimbra (PT). The questionnaire included socio-demographic characteristics and the amount of time spent in VPA. Results: The mean time spent practicing VPA was 60.4 min per day, with lower values for Namur (BE) and Latina (IT), and higher values for Amersfoort (NL). In the multivariable analysis, both categories of SEP indicators (family-based and adolescent based indicators) were independently associated with VPA. For each SEP indicator, lower levels of VPA were recorded in lower socioeconomic groups. In the total sample, each additional category of low SEP was associated with a decrease in mean VPA of about 4 min per day. Conclusions: This study showed that across European cities adolescent VPA is positively related to both family-based SEP and adolescents’ own SEP. When analysing socioeconomic differences in adolescent VPA, one should consider the use of multiple indicators of SEP.publishersversionpublishe
An Introduction to Conformal Ricci Flow
We introduce a variation of the classical Ricci flow equation that modifies
the unit volume constraint of that equation to a scalar curvature constraint.
The resulting equations are named the Conformal Ricci Flow Equations because of
the role that conformal geometry plays in constraining the scalar curvature.
These equations are analogous to the incompressible Navier-Stokes equations of
fluid mechanics inasmuch as a conformal pressure arises as a Lagrange
multiplier to conformally deform the metric flow so as to maintain the scalar
curvature constraint. The equilibrium points are Einstein metrics with a
negative Einstein constant and the conformal pressue is shown to be zero at an
equilibrium point and strictly positive otherwise. The geometry of the
conformal Ricci flow is discussed as well as the remarkable analytic fact that
the constraint force does not lose derivatives and thus analytically the
conformal Ricci equation is a bounded perturbation of the classical
unnormalized Ricci equation. That the constraint force does not lose
derivatives is exactly analogous to the fact that the real physical pressure
force that occurs in the Navier-Stokes equations is a bounded function of the
velocity. Using a nonlinear Trotter product formula, existence and uniqueness
of solutions to the conformal Ricci flow equations is proven. Lastly, we
discuss potential applications to Perelman's proposed implementation of
Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur
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