Conformal actions of ${\mathfrak{sl}_n(\mathbb{R})}$ and ${\hbox{SL}_n(\mathbb{R})\ltimes\mathbb{R}^n}$ on Lorentz manifolds
Scot
Adams;
Garrett
Stuck
3913-3936
Abstract: We prove that, for $n\ge3$, a locally faithful action of ${\hbox{SL}_n(\mathbb{R} )\ltimes\mathbb{R} ^n}$ or of $\hbox{SL}_n({\mathbb R})$ by conformal transformations of a connected Lorentz manifold must be a proper action.
Local boundary rigidity of a compact Riemannian manifold with curvature bounded above
Christopher
B.
Croke;
Nurlan
S.
Dairbekov;
Vladimir
A.
Sharafutdinov
3937-3956
Abstract: This paper considers the boundary rigidity problem for a compact convex Riemannian manifold $(M,g)$ with boundary $\partial M$whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics $g'$ on $M$ there is a $C^{3,\alpha }$-neighborhood of $g$ such that $g$is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in $M$). More precisely, given any metric $g'$ in this neighborhood with the same boundary distance function there is diffeomorphism $\varphi$which is the identity on $\partial M$such that
Some properties of minimal surfaces in singular spaces
Chikako
Mese
3957-3969
Abstract: This paper involves the generalization of minimal surface theory to spaces with singularities. Let $X$ be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in $X$ as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in $X$.
Uniform densities of regular sequences in the unit disk
Peter
L.
Duren;
Alexander
P.
Schuster;
Kristian
Seip
3971-3980
Abstract: The upper and lower uniform densities of some regular sequences are computed. These densities are used to determine sequences of sampling and interpolation for Bergman spaces of the unit disk.
The nonstationary ideal and the other $\sigma $-ideals on $\omega _{1}$
Jindrich
Zapletal
3981-3993
Abstract: Under Martin's Maximum every $\sigma$-ideal on $\omega _{1}$ is a subset of an ideal Rudin-Keisler reducible to a finite Fubini power of the nonstationary ideal restricted to a positive set.
Analytic types of plane curve singularities defined by weighted homogeneous polynomials
Chunghyuk
Kang
3995-4006
Abstract: We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials $f(y,z)$, which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that $g(y,z)$ either satisfies the same property as the above $f$ does or is homogeneous, then we prove easily that the weights of the above $g$ determine the topological type of $g$ and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in $\mathbb{C}^{2}$, which was already known. Also, as an application, it can be shown that for a given $h$, where $h(w_{1},\dots ,w_{n})$ is a quasihomogeneous holomorphic function with an isolated singularity at the origin or $h(w_{1})=w^{p}_{1}$ with a positive integer $p$, analytic types of isolated hypersurface singularities defined by $f+h$ are easily classified where $f$ is defined just as above.
Dihedral coverings of algebraic surfaces and their application
Hiro-o
Tokunaga
4007-4017
Abstract: In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following: Let $B$ be a reduced plane curve of even degree $d$ having only $a$ nodes and $b$ cusps. If $2a + 6b > 2d^2 - 6d + 6$, then $\pi_1(\mathbf{P}^2 \setminus B)$ is non-abelian. Note that Nori's result implies that $\pi_1(\mathbf{P}^2 \setminus B)$ is abelian, provided that $2a + 6b < d^2$.
Residues of a Pfaff system relative to an invariant subscheme
F.
Sancho de Salas
4019-4035
Abstract: In this paper we give a purely algebraic construction of the theory of residues of a Pfaff system relative to an invariant subscheme. This construction is valid over an arbitrary base scheme of any characteristic.
Linear systems of plane curves with base points of equal multiplicity
Ciro
Ciliberto;
Rick
Miranda
4037-4050
Abstract: In this article we address the problem of computing the dimension of the space of plane curves of degree $d$with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all $m \leq 12$.
A condition for the stability of $\mathbb{R}$-covered on foliations of 3-manifolds
Sue
Goodman;
Sandi
Shields
4051-4065
Abstract: We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be $\mathbb{R}$-covered. This condition can be readily verified for many examples. Further, if an $\mathbb{R}$-covered foliation has a compact leaf $L$, then any transverse loop meeting $L$ lifts to a copy of the leaf space, and the ambient manifold fibers over $S^1$ with $L$ as fiber.
The Markov spectra for Fuchsian groups
L.
Ya.
Vulakh
4067-4094
Abstract: Applying the Klein model $D^2$ of the hyperbolic plane and identifying the geodesics in $D^2$ with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.
A regular space with a countable network and different dimensions
George
Delistathis;
Stephen
Watson
4095-4111
Abstract: In this paper, we construct a regular space with a countable network (even the union of countably many separable metric subspaces) in which $ind$ and $dim$ do not coincide under the assumption of the continuum hypothesis (CH). This gives a consistent negative answer to a question of A.V. Arhangel'skii.
On the homotopy of simplicial algebras over an operad
Benoit
Fresse
4113-4141
Abstract: According to a result of H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra.
$p$-central groups and Poincaré duality
Thomas
S.
Weigel
4143-4154
Abstract: In this note we investigate the mod $p$ cohomology ring of finite $p$-central groups with a certain extension property. For $p$ odd it turns out that the structure of the cohomology ring characterizes this class of groups up to extensions by $p'$-groups. For certain examples the cohomology ring can be calculated explicitly. As a by-product one gets an alternative proof of a theorem of M.Lazard which states that the Galois cohomology of a uniformly powerful pro-$p$-group of rank $n$ is isomorphic to $\Lambda [x_{1},..,x_{n}]$.
Representing nonnegative homology classes of $\mathbb{C}P^2\#n\overline{\mathbb{C}P}{}^2$ by minimal genus smooth embeddings
Bang-He
Li
4155-4169
Abstract: For any nonnegative class $\xi$ in $H_2({\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2, {\mathbf Z})$, the minimal genus of smoothly embedded surfaces which represent $\xi$ is given for $n\leq 9$, and in some cases with $n\geq 10$, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus $g$, we prove that it is true for $n\leq 8$ with $g\geq 1$ and for $n\leq 9$ with $g\geq 2$.
The Conley index over a base
Marian
Mrozek;
James
F.
Reineck;
Roman
Srzednicki
4171-4194
Abstract: We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in ${R}^3$, and the problem of continuing two periodic orbits which are not homotopic as loops.
Resonance problems with respect to the Fucík spectrum
Martin
Schechter
4195-4205
Abstract: We study semilinear boundary value problems which have asymptotic resonance with respect to the linear part. The difficulties for Fucík resonance problems are compounded by the fact that there is no eigenspace with which to work. The present paper uses new linking theorems which can deal with the sets required to obtain critical points.
Robin boundary value problems on arbitrary domains
Daniel
Daners
4207-4236
Abstract: We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish $L_p$-$L_q$-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.
A Palais-Smale approach to problems in Esteban-Lions domains with holes
Hwai-Chiuan
F.
Wang
4237-4256
Abstract: Let $\Omega \subset {\mathbb{R} }^{N}$ be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in $\Omega$ and describe the dynamic systems of solutions of equation $(1)$ in various $\Omega$. We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in $\Omega$.
A sampling theorem on homogeneous manifolds
Isaac
Pesenson
4257-4269
Abstract: We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.
Estimates for functions of the Laplace operator on homogeneous trees
Michael
Cowling;
Stefano
Meda;
Alberto
G.
Setti
4271-4293
Abstract: In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest-neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the $L^{p}-L^{q}$ mapping properties of the corresponding operators.
The Noetherian property in some quadratic algebras
Xenia
H.
Kramer
4295-4323
Abstract: We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph $\Gamma (A)$. The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph $\Gamma (A)$ to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.
The $q$-Schur${}^{2}$ algebra
Jie
Du;
Leonard
Scott
4325-4353
Abstract: We study a class of endomomorphism algebras of certain $q$-permutation modules over the Hecke algebra of type $B$, whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed.
Specializations of Brauer classes over algebraic function fields
Burton
Fein;
Murray
Schacher
4355-4369
Abstract: Let $F$ be either a number field or a field finitely generated of transcendence degree $\ge 1$ over a Hilbertian field of characteristic 0, let $F(t)$ be the rational function field in one variable over $F$, and let ${\alpha }\in \operatorname {Br}(F(t))$. It is known that there exist infinitely many $a\in F$ such that the specialization $t\to a$ induces a specialization ${\alpha }\to \overline {{\alpha }}\in \operatorname {Br}(F)$, where $\overline {{\alpha }}$ has exponent equal to that of ${\alpha }$. Now let $K$ be a finite extension of $F(t)$ and let ${\beta }=\operatorname {res}_{K/F(t)}({\alpha })$. We give sufficient conditions on ${\alpha }$ and $K$ for there to exist infinitely many $a\in F$ such that the specialization $t\to a$has an extension to $K$ inducing a specialization ${\beta }\to \overline {{\beta }}\in \operatorname {Br}(\overline{K})$, $\overline{K}$ the residue field of $K$, where $\overline {{\beta }}$ has exponent equal to that of ${\beta }$. We also give examples to show that, in general, such $a\in F$ need not exist.
Conformally invariant Monge-Ampère equations: Global solutions
Jeff
A.
Viaclovsky
4371-4379
Abstract: In this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group $SO(n+1,1)$. These equations are elliptic and variational. Using this structure and the conformal invariance, we will prove a global uniqueness theorem for solutions in $\mathbf{R}^n$ with a quadratic growth condition at infinity.
Geometric properties of the sections of solutions to the Monge-Ampère equation
Cristian
E.
Gutiérrez;
Qingbo
Huang
4381-4396
Abstract: In this paper we establish several geometric properties of the cross sections of generalized solutions $\phi$ to the Monge-Ampère equation $\det D^{2}\phi = \mu$, when the measure $\mu$ satisfies a doubling property. A main result is a characterization of the doubling measures $\mu$in terms of a geometric property of the cross sections of $\phi$. This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.