86 research outputs found
General existence proof for rest frame systems in asymptotically flat space-time
We report a new result on the nice section construction used in the
definition of rest frame systems in general relativity. This construction is
needed in the study of non trivial gravitational radiating systems. We prove
existence, regularity and non-self-crossing property of solutions of the nice
section equation for general asymptotically flat space times. This proves a
conjecture enunciated in a previous work.Comment: 14 pages, no figures, LaTeX 2
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles
The construction of sections of bundles with prescribed jet values plays a
fundamental role in problems of algebraic and complex geometry. When the jet
values are prescribed on a positive dimensional subvariety, it is handled by
theorems of Ohsawa-Takegoshi type which give extension of line bundle valued
square-integrable top-degree holomorphic forms from the fiber at the origin of
a family of complex manifolds over the open unit 1-disk when the curvature of
the metric of line bundle is semipositive. We prove here an extension result
when the curvature of the line bundle is only semipositive on each fiber with
negativity on the total space assumed bounded from below and the connection of
the metric locally bounded, if a square-integrable extension is known to be
possible over a double point at the origin. It is a Hensel-lemma-type result
analogous to Artin's application of the generalized implicit function theorem
to the theory of obstruction in deformation theory. The motivation is the need
in the abundance conjecture to construct pluricanonical sections from flatly
twisted pluricanonical sections. We also give here a new approach to the
original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the
punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi
to a simple application of the standard method of constructing holomorphic
functions by solving the d-bar equation with cut-off functions and additional
blowup weight functions
Characterization of Generalized Young Measures Generated by Symmetric Gradients
This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer\ue2\u80\u93Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The \ue2\u80\u9clocal\ue2\u80\u9d proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti\ue2\u80\u99s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences
The masterpieces of John Forbes Nash Jr.
In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader
Existence and Partial Regularity in the Calculus of Variations
We prove existence and partial regularity of minimizers of certain functionals in the calculus of variations, under the main assumption that integrands be quasiconvex, and introduce a definition of general-quasiminima (G - Qminima), and prove some regularity results in Lfor G - Qminima as well as for Q-minima
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