60 research outputs found
On the local extension of Killing vector-fields in Ricci flat manifolds
We revisit the problem of extension of Killing vector-fields in smooth Ricci
flat manifolds, and its relevance to the black hole rigidity problem
A fully anisotropic mechanism for formation of trapped surfaces in vacuum
We present a new, fully anisotropic, criterion for formation of trapped
surfaces in vacuum. More precisely we provide conditions on null data,
concentrated in a neighborhood of a short null geodesic segment (possibly flat
everywhere else) whose future development contains a trapped surface. This
extends considerably the previous result of Christodoulou \cite{Chr:book} which
required instead a uniform condition along all null geodesic generators. To
obtain our result we combine Christodoulou's mechanism for the formation of a
trapped surface with a new deformation process which takes place along incoming
null hypersurfaces
On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
We give a new proof of uniqueness of solutions to the Gross-Pitaevskii
hierarchy, first established by Erdos, Schlein and Yau, in a different space,
based on space-time estimates
The Bounded L2 Curvature Conjecture
This is the main paper in a sequence in which we give a complete proof of the
bounded curvature conjecture. More precisely we show that the time of
existence of a classical solution to the Einstein-vacuum equations depends only
on the -norm of the curvature and a lower bound on the volume radius of
the corresponding initial data set. We note that though the result is not
optimal with respect to the standard scaling of the Einstein equations, it is
nevertheless critical with respect to its causal geometry. Indeed, bounds
on the curvature is the minimum requirement necessary to obtain lower bounds on
the radius of injectivity of causal boundaries. We note also that, while the
first nontrivial improvements for well posedness for quasilinear hyperbolic
systems in spacetime dimensions greater than 1+1 (based on Strichartz
estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized
in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full
structure of the quasilinear hyperbolic system, not just its principal part,
plays a crucial role. To achieve our goals we recast the Einstein vacuum
equations as a quasilinear -valued Yang-Mills theory and introduce a
Coulomb type gauge condition in which the equations exhibit a specific new type
of \textit{null structure} compatible with the quasilinear, covariant nature of
the equations. To prove the conjecture we formulate and establish bilinear and
trilinear estimates on rough backgrounds which allow us to make use of that
crucial structure. These require a careful construction and control of
parametrices including error bounds which is carried out in [Sz1]-[Sz4],
as well as a proof of sharp Strichartz estimates for the wave equation on a
rough background which is carried out in \cite{Sz5}.Comment: updated version taking into account the remarks of the refere
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