Volume 2014, Issue 690

Abstract. The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painlevé equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the canonical variables.

Abstract. We study vector-valued p -adic Siegel modular forms based on the arithmetic theory of vector-valued Siegel modular forms over rings. We construct Fourier expansions and Hecke operators for these modular forms, and especially, using the description of these p -adic forms on the Igusa tower, we give a natural definition of p -adic operators.

Abstract. We present a proof of a Littlewood–Richardson rule for the K -theory of odd orthogonal Grassmannians OG( n ,2 n +1), as conjectured by Thomas–Yong (2009). Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, Buch–Ravikumar (2012) proved a Pieri rule for OG( n ,2 n +1) that confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.

Abstract. We consider the energy critical nonlinear Schrödinger equation on periodic domains of the form ℝm×𝕋4-m$\mathbb {R}^m\times \mathbb {T}^{4-m}$ with m=0,1,2,3$m = 0,1,2,3$. Assuming that a certain L 4 Strichartz estimate holds for solutions to the corresponding linear Schrödinger equation, we prove that the nonlinear problem is locally well-posed in the energy space. Then we verify that the L 4 estimate holds if m=2,3$m=2,3$.

Abstract. We present a proof of a Minami type estimate for one-dimensional random Schrödinger operators valid at all energies in the localization regime provided a Wegner estimate is known to hold. The Minami type estimate is then applied to two models to obtain results on their spectral statistics. The heuristics underlying our proof of Minami type estimates is that close-by eigenvalues of a one-dimensional Schrödinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena. In the second case, one can undo the tunneling and thus construct quasi-modes that live far away from each other in space.

Abstract. We study the mean curvature flow of graphs with prescribed contact angle on a fixed, smooth hyperplane in Euclidean space. We obtain long time existence and convergence to a self similar solution of the mean curvature flow orthogonal to the fixed hyperplane.

Abstract. We prove that the reduced holonomy group of a complete smooth solution to the Ricci flow of uniformly bounded curvature cannot spontaneously contract within the lifetime of the solution. It follows then, from an earlier result of Hamilton, that the holonomy is exactly preserved by the equation. In particular, a solution to the Ricci flow may be Kähler or locally reducible at t=T$t= T$ if and only if the same is true of g(t)$g(t)$ at times t≤T$t\le T$.

Abstract. We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings. Furthermore, given such a closed, multiplicity-one, smoothly embedded self-similar shrinker Σ, we show that any solution of the rescaled flow, which is sufficiently close to Σ, with Gaussian density ratios greater or equal to that of Σ, stays for all time close to Σ and converges to a possibly different self-similarly shrinking solution Σ'$\Sigma ^{\prime }$. The central point in the argument is a direct application of the Łojasiewicz–Simon inequality to Huisken's monotone Gaussian integral for Mean Curvature Flow.

Abstract. We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.

Abstract. In this article, we classify the characters associated to algebraic points on Shimura curves of Γ 0 ( p )-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number p . This is an analogue of the study of rational points or points over a quadratic field on the modular curve X0(p)$X_0(p)$ by Mazur and the second author. We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen–Tamagawa.

Abstract. This is an erratum to http://dx.doi.org/10.1515/crelle-2012-0068.

Abstract. We study property A defined by G. Yu and the operator norm localization property defined by X. Chen, R. Tessera, X. Wang, and G. Yu. These are coarse geometric properties for metric spaces, which have applications to operator K-theory. It is proved that the two properties are equivalent for discrete metric spaces with bounded geometry.

Abstract. We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in ℂ d to be biholomorphic to ℂ d . As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson–Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are well-defined is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of the conjugacy operator by a lower triangular matrix on the space of k -homogeneous polynomial endomorphisms of ℂ d .