Sasakian-Einstein structures on $9\#(S^2\times S^3)$
Charles
P.
Boyer;
Krzysztof
Galicki;
Michael
Nakamaye
2983-2996
Abstract: We show that $\scriptstyle{9\char93 (S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.
The Bergman metric on a Stein manifold with a bounded plurisubharmonic function
Bo-Yong
Chen;
Jin-Hao
Zhang
2997-3009
Abstract: In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature $\le -A/\rho^2$, which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.
Irreducibility, Brill-Noether loci, and Vojta's inequality
Thomas
J.
Tucker;
with an Appendix by Olivier
Debarre
3011-3029
Abstract: This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover $f$ of the projective line defined over a number field $k$, there exist infinitely many $k$-rational points on the projective line such that the fiber of $f$ over $P$is irreducible over $k$. In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.
Birational automorphisms of quartic Hessian surfaces
Igor
Dolgachev;
JongHae
Keum
3031-3057
Abstract: We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice $II_{1,25}$ of rank 26 and signature $(1,25)$. The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.
Light structures in infinite planar graphs without the strong isoperimetric property
Bojan
Mohar
3059-3074
Abstract: It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree $\le 5$ and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles. Every planar graph with isoperimetric number zero and with essential minimum degree $\ge3$ has infinitely many edges whose degree sum is at most 15. In particular, this holds for all graphs with minimum degree $\ge3$ and with subexponential growth. The cases without infinitely many edges whose degree sum is $\le14$ (or, similarly, $\le13$ or $\le 12$) are also considered. Several further results are obtained.
Algebraic structure in the loop space homology Bockstein spectral sequence
Jonathan
A.
Scott
3075-3084
Abstract: Let $X$ be a finite, $n$-dimensional, $r$-connected CW complex. We prove the following theorem: If $p \geq n/r$ is an odd prime, then the loop space homology Bockstein spectral sequence modulo $p$ is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.
On certain co--H spaces related to Moore spaces
Manfred
Stelzer
3085-3093
Abstract: We show that certain co-$H$ spaces, constructed by Anick and Gray, carry a homotopy co-associative and co-commutative co-$H$ structure.
Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves
L.
Caffarelli;
J.
Salazar
3095-3115
Abstract: In this paper, we first construct ``viscosity'' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form \begin{displaymath}F(D^{2} u,x) = g(x,u) \text{ on } \{\vert\nabla u\vert \ne 0\}\end{displaymath} In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test'' polynomials $P$ (those tangent from above or below to the graph of $u$ at a point $x_{0}$) satisfy the correct inequality only if $\vert\nabla P (x_{0})\vert \ne 0$. That is, we simply disregard those test polynomials for which $\vert\nabla P (x_{0})\vert = 0$. Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for $F = \Delta$, $g = cu$($c>0$) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.
Semilinear Neumann boundary value problems on a rectangle
Junping
Shi
3117-3154
Abstract: We consider a semilinear elliptic equation \begin{displaymath}\Delta u+\lambda f(u)=0, \;\; \mathbf{x}\in \Omega,\;\; \frac{\partial u}{\partial n }=0, \;\; {\mathbf x}\in \partial \Omega, \end{displaymath} where $\Omega$ is a rectangle $(0,a)\times(0,b)$ in $\mathbf{R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda\to\infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.
Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems
De-xing
Kong
3155-3179
Abstract: Employing the method of characteristic coordinates and the singularity theory of smooth mappings, in this paper we analyze the long-term behaviour of smooth solutions of general $2\times 2$ quasilinear hyperbolic systems, provide a complete description of the solution close to blow-up points, and investigate the formation and propagation of singularities for $2\times 2$ systems of hyperbolic conservation laws.
Compactness of the solution operator for a linear evolution equation with distributed measures
Ioan
I.
Vrabie
3181-3205
Abstract: The main goal of the present paper is to define the solution operator $(\xi,g)\mapsto u$associated to the evolution equation $du=(Au)dt+dg$, $u(0)=\xi$, where $A$generates a $C_0$-semigroup in a Banach space $X$, $\xi\in X$, $g\in BV([\,a,b\,];X)$, and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to BARAS, HASSAN, VERON, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.
Ljusternik-Schnirelman theory in partially ordered Hilbert spaces
Shujie
Li;
Zhi-Qiang
Wang
3207-3227
Abstract: We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.
Transfer functions of regular linear systems Part II: The system operator and the Lax--Phillips semigroup
Olof
Staffans;
George
Weiss
3229-3262
Abstract: This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as ``Part I''. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by $\dot x=Ax+Bu$, $y=Cx+Du$ would be the $s$-dependent matrix $S_\Sigma(s)= \left[ {}^{A-sI}_{ \;\,C} { } ^{B}_{D} \right]$. In the general case, $S_\Sigma(s)$ is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks $A-sI$ and $B$, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of $S_\Sigma(s)$ where the right lower block is the feedthrough operator of the system. Using $S_\Sigma(0)$, we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the ``initial time'' is $-\infty$. We also introduce the Lax-Phillips semigroup $\boldsymbol{\mathfrak{T}}$ induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ${\omega}\in{\mathbb R}$which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of $A$ and also the points where $S_\Sigma(s)$ is not invertible, in terms of the spectrum of the generator of $\boldsymbol{\mathfrak{T}}$(for various values of ${\omega}$). The system $\Sigma$is dissipative if and only if $\boldsymbol{\mathfrak{T}}$(with index zero) is a contraction semigroup.
On the centered Hardy-Littlewood maximal operator
Antonios
D.
Melas
3263-3273
Abstract: We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant $C$ in the $L^{1}\rightarrow$ weak $L^{1}$ inequality for this operator. In fact we will show that $\frac{11+\sqrt{61}}{12}=1.5675208...\leq C\leq\frac{5} {3}=1.66...$.
On a class of jointly hyponormal Toeplitz operators
Caixing
Gu
3275-3298
Abstract: We characterize when a pair of Toeplitz operators $\mathbf{T}=(T_{\phi },T_{\psi})$ is jointly hyponormal under various assumptions--for example, $\phi$ is analytic or $\phi$ is a trigonometric polynomial or $\phi-\psi$ is analytic. A typical characterization states that $\mathbf{T}=(T_{\phi },T_{\psi})$ is jointly hyponormal if and only if an algebraic relation of $\phi$ and $\psi$ holds and the single Toeplitz operator $T_{\omega}$ is hyponormal, where $\omega$ is a combination of $\phi$ and $\psi$. More general results for an $n$-tuple of Toeplitz operators are also obtained.
Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces
Simon
Gindikin;
Bernhard
Krötz
3299-3327
Abstract: In this paper we define a distinguished boundary for the complex crowns $\Xi \subseteq G_{\mathbb{C} } /K_{\mathbb{C} }$ of non-compact Riemannian symmetric spaces $G/K$. The basic result is that affine symmetric spaces of $G$ can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.
Solvable groups with polynomial Dehn functions
G.
N.
Arzhantseva;
D.
V.
Osin
3329-3348
Abstract: Given a finitely presented group $H$, finitely generated subgroup $B$ of $H$, and a monomorphism $\psi :B\to H$, we obtain an upper bound of the Dehn function of the corresponding HNN-extension $G=\langle H, t\; \vert\; t^{-1}Bt=\psi (B)\rangle$ in terms of the Dehn function of $H$ and the distortion of $B$ in $G$. Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol'shanskii, Rips, and Sapir.
Hopf modules and the double of a quasi-Hopf algebra
Peter
Schauenburg
3349-3378
Abstract: We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.
Invariant ideals and polynomial forms
D.
S.
Passman
3379-3408
Abstract: Let $K[\mathfrak H]$ denote the group algebra of an infinite locally finite group $\mathfrak H$. In recent years, the lattice of ideals of $K[\mathfrak H]$has been extensively studied under the assumption that $\mathfrak H$ is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra $K[V]$stable under the action of an appropriate automorphism group of $V$. Specifically, in this paper, we let ${\mathfrak{G}}$ be a quasi-simple group of Lie type defined over an infinite locally finite field $F$, and we let $V$ be a finite-dimensional vector space over a field $E$ of the same characteristic $p$. If ${\mathfrak{G}}$ acts nontrivially on $V$ by way of the homomorphism $\phi\colon{\mathfrak{G}}\to\mathrm{GL}(V)$, and if $V$ has no proper ${\mathfrak{G}}$-stable subgroups, then we show that the augmentation ideal $\omega K[V]$ is the unique proper ${\mathfrak{G}}$-stable ideal of $K[V]$ when ${\operatorname{char}} K\neq p$. The proof of this result requires, among other things, that we study characteristic $p$ division rings $D$, certain multiplicative subgroups $G$ of $D^{\bullet}$, and the action of $G$ on the group algebra $K[A]$, where $A$ is the additive group $D^{+}$. In particular, properties of the quasi-simple group ${\mathfrak{G}}$ come into play only in the final section of this paper.