10,951 research outputs found
Pinched hypersurfaces contract to round points
We investigate the evolution of closed strictly convex hypersurfaces in
, n=3, for contracting normal velocities, including powers of
the mean curvature, of the norm of the second fundamental form, and of the
Gauss curvature. We prove convergence to a round point for 2-pinched initial
hypersurfaces. In , n=2, natural quantities exist for proving
convergence to a round point for many normal velocities. Here we present their
counterparts for arbitrary dimensions .Comment: 14 page
On maximum-principle functions for flows by powers of the Gauss curvature
We consider flows with normal velocities equal to powers strictly larger than
one of the Gauss curvature. Under such flows closed strictly convex surfaces
converge to points. In his work on the square of the norm of the second
fundamental form, Schn\"urer proposes criteria for selecting quantities that
are suitable for proving convergence to a round point. Such monotone quantities
exist for many normal velocities, including the Gauss curvature, some powers
larger than one of the mean curvature, and some powers larger than one of the
norm of the second fundamental form. In this paper, we show that no such
quantity exists for any powers larger than one of the Gauss curvature.Comment: 31 page
Boundedness of massless scalar waves on Reissner-Nordstr\"om interior backgrounds
We consider solutions of the scalar wave equation , without
symmetry, on fixed subextremal Reissner-Nordstr\"om backgrounds with nonvanishing charge. Previously, it has been shown that for
arising from sufficiently regular data on a two ended Cauchy hypersurface, the
solution and its derivatives decay suitably fast on the event horizon
. Using this, we show here that is in fact uniformly
bounded, , in the black hole interior up to and including the
bifurcate Cauchy horizon , to which in fact
extends continuously. The proof depends on novel weighted energy estimates in
the black hole interior which, in combination with commutation by angular
momentum operators and application of Sobolev embedding, yield uniform
pointwise estimates. In a forthcoming companion paper we will extend the result
to subextremal Kerr backgrounds with nonvanishing rotation.Comment: minor improvements, references adde
The P versus NP Brief
This paper discusses why P and NP are likely to be different. It analyses the
essence of the concepts and points out that P and NP might be diverse by sheer
definition. It also speculates that P and NP may be unequal due to natural
laws.Comment: 4 pages, 1 figure; added notational definition for functions for
section 2, formatting and wording changes; corrected typo, recompiled
pdf-fil
On Cohomology Rings of Non-Commutative Hilbert Schemes and CoHa-Modules
We prove that Chow groups of certain non-commutative Hilbert schemes have a
basis consisting of monomials in Chern classes of the universal bundle.
Furthermore, we realize the cohomology of non-commutative Hilbert schemes as a
module over the Cohomological Hall algebra.Comment: 28 pages. v2: Final version; to appear in Math. Res. Let. Improved
exposition in subsection 2.2 (thanks to the referee), results in section 3
hold for an arbitrary framing datum
Entire Graphs Evolving by Powers of the Mean Curvature
We study convex entire graphs evolving with normal velocity equal to a
positive power of the mean curvature. Under mild assumptions we prove longtime
existence.Comment: 14 pages, 2 figures. arXiv admin note: text overlap with
arXiv:math/0612659 by different author
When maximum-principle functions cease to exist
We consider geometric flow equations for contracting and expanding normal
velocities, including powers of the Gauss curvature, of the mean curvature, and
of the norm of the second fundamental form, and ask whether - after appropriate
rescaling - closed strictly convex surfaces converge to spheres. To prove this,
many authors use certain functions of the principal curvatures, which we call
maximum-principle functions. We show when such functions cease to exist and
exist, while presenting newly discovered maximum-principle functions.Comment: 49 pages, 1 figure, 7 table
Chow Rings of Fine Quiver Moduli are Tautologically Presented
A result of A. King and C. Walter asserts that the Chow ring of a fine quiver
moduli space is generated by the Chern classes of universal bundles if the
quiver is acyclic. We will show that defining relations between these Chern
classes arise geometrically as degeneracy loci associated to the universal
representation.Comment: 30 pages, final version to appear in Math.
Semi-Stable Chow-Hall Algebras of Quivers and Quantized Donaldson-Thomas Invariants
The semi-stable ChowHa of a quiver with stability is defined as an analog of
the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in
equivariant Chow groups of semi-stable loci in representation varieties of
quivers. We prove several structural results on the semi-stable ChowHa, namely
isomorphism of the cycle map, a tensor product decomposition, and a
tautological presentation. For symmetric quivers, this leads to an
identification of their quantized Donaldson-Thomas invariants with the
Chow-Betti numbers of moduli spaces.Comment: 23 pages; v2: Fixed an error in the proof of Thm. 5.1, generalized
Thm. 6.1 to arbitrary quivers (not just symmetric ones
Non-Schurian indecomposables via intersection theory
For an acyclic quiver with three vertices, we consider the canonical
decomposition of a non-Schurian root and associate certain representations of a
generalized Kronecker quiver. These representations correspond to points
contained in the intersection of two subvarieties of a Grassmannian and give
rise to representations of the original quiver, preserving indecomposability.
We show that these subvarieties intersect using Schubert calculus. Provided
that the intersection contains a Schurian representation, it already contains
an open subset of Schurian representations whose dimension is what we expect by
Kac's Theorem.Comment: 38 pages; v2: introduction rewritten, added recollection on Ringel's
reflection functor (subsect. 2.3), improved exposition in sect.
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