5,354 research outputs found
Topological censorship from the initial data point of view
We introduce a natural generalization of marginally outer trapped surfaces,
called immersed marginally outer trapped surfaces, and prove that three
dimensional asymptotically flat initial data sets either contain such surfaces
or are diffeomorphic to R^3. We establish a generalization of the Penrose
singularity theorem which shows that the presence of an immersed marginally
outer trapped surface generically implies the null geodesic incompleteness of
any spacetime that satisfies the null energy condition and which admits a
non-compact Cauchy surface. Taken together, these results can be viewed as an
initial data version of the Gannon-Lee singularity theorem. The first result is
a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that
every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to
R^3 contains an embedded stable minimal surface. We also obtain an initial data
version of the spacetime principle of topological censorship. Under physically
natural assumptions, a 3-dimensional asymptotically flat initial data set with
marginally outer trapped boundary and no immersed marginally outer trapped
surfaces in its interior is diffeomorphic to R^3 minus a finite number of open
balls. An extension to higher dimensions is also discussed.Comment: v2: Appendix added, Theorem 5.1 improved, other minor changes. To
appear in J. Diff. Geo
Lefschetz fibrations and Torelli groups
For each g > 2 and h > 1, we explicitly construct (1) fiber sum
indecomposable relatively minimal genus g Lefschetz fibrations over genus h
surfaces whose monodromies lie in the Torelli group, (2) fiber sum
indecomposable genus g surface bundles over genus h surfaces whose monodromies
are in the Torelli group (provided g > 3), and (3) infinitely many genus g
Lefschetz fibrations over genus h surfaces that are not fiber sums of
holomorphic ones.Comment: 20 pages, 3 figure
Boring split links
Boring is an operation which converts a knot or two-component link in a
3--manifold into another knot or two-component link. It generalizes rational
tangle replacement and can be described as a type of 2--handle attachment.
Sutured manifold theory is used to study the existence of essential spheres and
planar surfaces in the exteriors of knots and links obtained by boring a split
link. It is shown, for example, that if the boring operation is complicated
enough, a split link or unknot cannot be obtained by boring a split link.
Particular attention is paid to rational tangle replacement. If a knot is
obtained by rational tangle replacement on a split link, and a few minor
conditions are satisfied, the number of boundary components of a meridional
planar surface is bounded below by a number depending on the distance of the
rational tangle replacement. This result is used to give new proofs of two
results of Eudave-Mu\~noz and Scharlemann's band sum theorem.Comment: 43 pages, 12 figures; minor changes and corrections. Accepted by
Pacific Journal of Mathematic
Topology of Cosmological Black Holes
Motivated by the question of how generic inflation is, I study the
time-evolution of topological surfaces in an inhomogeneous cosmology with
positive cosmological constant . If matter fields satisfy the Weak
Energy Condition, non-spherical incompressible surfaces of least area are shown
to expand at least exponentially, with rate , under the mean curvature flow parametrized by . With
reasonable assumptions about the nature of singularities this restricts the
topology of black holes: (a) no trapped surface or apparent horizon can be a
non-spherical, incompressible surface, and (b) the interior of black holes
cannot contain any such surface.Comment: JCAP version, 30 pages, 6 figures, our definition of black holes and
apparent horizons in cosmology has been clarifie
The Weyl Law for the phase transition spectrum and density of limit interfaces
We prove a Weyl Law for the phase transition spectrum based on the techniques
of Liokumovich-Marques-Neves. As an application we give phase transition
adaptations of the proofs of the density and equidistribution of minimal
hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song,
respectively. We also prove the density of separating limit interfaces for
generic metrics in dimension 3, based on the recent work of
Chodosh-Mantoulidis, and for generic metrics on manifolds containing only
separating minimal hypersurfaces, e.g. , for . These provide alternative proofs of Yau's conjecture on the
existence of infinitely many minimal hypersurfaces for generic metrics on each
setting, using the Allen-Cahn approach.Comment: 26 pages, comments welcome. v2: expanded and detailed proof of the
last result, references update
Rigidity of outermost MOTS - the initial data version
In [5], a rigidity result was obtained for outermost marginally outer trapped
surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This
allowed one to treat the "borderline case" in the author's work with R. Schoen
concerning the topology of higher dimensional black holes [8]. The proof of
this rigidity result involved bending the initial data manifold in the vicinity
of the MOTS within the ambient spacetime. In this note we show how to
circumvent this step, and thereby obtain a pure initial data version of this
rigidity result and its consequence concerning the topology of black holes.Comment: 8 pages; v2: minor changes; version to appear in GR
A note on the connectivity of certain complexes associated to surfaces
This note is devoted to a trick which yields almost trivial proofs that
certain complexes associated to topological surfaces are connected or simply
connected. Applications include new proofs that the complexes of curves,
separating curves, nonseparating curves, pants, and cut systems are all
connected for genus . We also prove that two new complexes are
connected : one involves curves which split a genus surface into two genus
pieces, and the other involves curves which are homologous to a fixed
curve. The connectivity of the latter complex can be interpreted as saying the
``homology'' relation on the surface is (for ) generated by
``embedded/disjoint homologies''. We finally prove that the complex of
separating curves is simply connected for .Comment: 15 pages, 2 figures, minor revisions; to appear in L'Enseignement
Mathematiqu
Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds
We show that the pre-order defined on the category of contact manifolds by
arbitrary symplectic cobordisms is considerably less rigid than its
counterparts for exact or Stein cobordisms: in particular, we exhibit large new
classes of contact 3-manifolds which are symplectically cobordant to something
overtwisted, or to the tight 3-sphere, or which admit symplectic caps
containing symplectically embedded spheres with vanishing self-intersection.
These constructions imply new and simplified proofs of several recent results
involving fillability, planarity and non-separating contact type embeddings.
The cobordisms are built from generalized symplectic handles which have cores
that are arbitrary symplectic surfaces with boundary and co-cores that are
symplectic disks or annuli; these can be attached to contact 3-manifolds along
sufficiently large neighborhoods of transverse links or pre-Lagrangian tori. We
also sketch a construction of J-holomorphic foliations in these cobordisms and
formulate a conjecture regarding maps induced on Embedded Contact Homology with
twisted coefficients.Comment: 50 pages, 7 figures; v.3 incorporates several corrections and changes
suggested by referees, including updated references and one (easy) new result
on the intersection forms of fillings of partially planar contact manifolds
(Theorem 10); to appear in J. Differential Geo
Torelli buildings and their automorphisms
In this paper we introduce, for each closed orientable surface, an analogue
of Tits buildings adjusted to investigation of the Torelli group of this
surface. It is a simplicial complex with some additional structure. We call
this complex with its additional structure the Torelli building of the surface
in question. The main result of this paper shows that Torelli buildings of
surfaces of genus at least 5 have only obvious automorphisms, and identifies
its group of automorphisms. Namely, we prove that for such a surface every
automorphism of its Torelli building is induced by a diffeomorphism of the
surface. This theorem about automorphisms of Torelli buildings is intended for
applications to automorphisms and virtual automorphisms of Torelli groups. The
latter results will be presented on some other occasion. All these results were
announced in arXiv:math/0311123.Comment: 39 pages, no figure
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