Pinched hypersurfaces contract to round points

We investigate the evolution of closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$, 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 $\mathbb{R}^{n+1}$, n=2, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\mathbb{N}$.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 $\Box_g\phi=0$, without symmetry, on fixed subextremal Reissner-Nordstr\"om backgrounds $({\mathcal M}, g)$ with nonvanishing charge. Previously, it has been shown that for $\phi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon ${\mathcal H}^+$. Using this, we show here that $\phi$ is in fact uniformly bounded, $|\phi| \leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon ${\mathcal C}{\mathcal H}^+$, to which $\phi$ 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.