2,376 research outputs found
Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds
Motivated by the geometric theory of differential equations and the
variational approach to the equivalence problem for geometric structures on
manifolds, we consider the problem of equivalence for distributions with fixed
submanifolds of flags on each fiber. We call them flag structures. The
construction of the canonical frames for these structures can be given in the
two prolongation steps: the first step, based on our previous works, gives the
canonical bundle of moving frames for the fixed submanifolds of flags on each
fiber and the second step consists of the prolongation of the bundle obtained
in the first step. The bundle obtained in the first step is not as a rule a
principal bundle so that the classical Tanaka prolongation procedure for
filtered structures can not be applied to it. However, under natural
assumptions on submanifolds of flags and on the ambient distribution, this
bundle satisfies a nice weaker property. The main goal of the present paper is
to formalize this property, introducing the so-called quasi-principle frame
bundles, and to generalize the Tanaka prolongation procedure to these bundles.
Applications to the equivalence problems for systems of differential equations
of mixed order, bracket generating distributions, sub-Riemannian and more
general structures on distributions are given.Comment: 49 pages. The Introduction was extended substantially: we demonstrate
how flag structures appear in the geometry of double fibrations and, using
this language, we discuss the motivating examples in more detai
Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
We rigorously define the Liouville action functional for finitely generated,
purely loxodromic quasi-Fuchsian group using homology and cohomology double
complexes naturally associated with the group action. We prove that the
classical action - the critical point of the Liouville action functional,
considered as a function on the quasi-Fuchsian deformation space, is an
antiderivative of a 1-form given by the difference of Fuchsian and
quasi-Fuchsian projective connections. This result can be considered as global
quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity.
We prove that the classical action is a Kahler potential of the Weil-Petersson
metric. We also prove that Liouville action functional satisfies holography
principle, i.e., it is a regularized limit of the hyperbolic volume of a
3-manifold associated with a quasi-Fuchsian group. We generalize these results
to a large class of Kleinian groups including finitely generated, purely
loxodromic Schottky and quasi-Fuchsian groups and their free combinations.Comment: 60 pages, proof of the Lemma 5.1 corrected, references and section
5.3 adde
Mean curvature flow and quasilocal mass for two-surfaces in Hamiltonian General Relativity
A family of quasilocal mass definitions that includes as special cases the
Hawking mass and the Brown-York ``rest mass'' energy is derived for spacelike
2-surfaces in spacetime. The definitions involve an integral of powers of the
norm of the spacetime mean curvature vector of the 2-surface, whose properties
are connected with apparent horizons. In particular, for any spacelike
2-surface, the direction of mean curvature is orthogonal (dual in the normal
space) to a unique normal direction in which the 2-surface has vanishing
expansion in spacetime. The quasilocal mass definitions are obtained by an
analysis of boundary terms arising in the gravitational ADM Hamiltonian on
hypersurfaces with a spacelike 2-surface boundary, using a geometric time-flow
chosen proportional to the dualized mean curvature vector field at the boundary
surface. A similar analysis is made choosing a geometric rotational flow given
in terms of the twist covector of the dual pair of mean curvature vector
fields, which leads to a family of quasilocal angular momentum definitions
involving the squared norm of the twist. The large sphere limit of these
definitions is shown to yield the ADM mass and angular momentum in
asymptotically flat spacetimes, while at apparent horizons a quasilocal version
of the Gibbons-Penrose inequality is derived. Finally, some results concerning
positivity are proved for the quasilocal masses, motivated by consideration of
spacelike mean curvature flow of 2-surfaces in spacetime.Comment: Revised version, includes an analysis of null flows with applications
to mass and angular momentum for apparent horizon
Causal Fermion Systems -- An Overview
The theory of causal fermion systems is an approach to describe fundamental
physics. We here introduce the mathematical framework and give an overview of
the objectives and current results.Comment: 54 pages, LaTeX, 1 figure, minor improvements (published version
Nonlinear Analysis and Optimization with Applications
Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world
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