Dimension of hyperbolic measures of random diffeomorphisms
Pei-Dong
Liu;
Jian-Sheng
Xie
3751-3780
Abstract: We consider dynamics of compositions of stationary random $C^2$ diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.
$3$-manifolds with planar presentations and the width of satellite knots
Martin
Scharlemann;
Jennifer
Schultens
3781-3805
Abstract: We consider compact $3$-manifolds $M$ having a submersion $h$ to $R$ in which each generic point inverse is a planar surface. The standard height function on a submanifold of $S^{3}$ is a motivating example. To $(M, h)$ we associate a connectivity graph $\Gamma$. For $M \subset S^{3}$, $\Gamma$ is a tree if and only if there is a Fox reimbedding of $M$ which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of $S^{3} - M$ is a tree, then there is a level-preserving reimbedding of $M$ so that $S^{3} - M$ is a connected sum of handlebodies. Corollary. $\bullet$ The width of a satellite knot is no less than the width of its pattern knot and so $\bullet$ $w(K_{1} \char93 K_{2}) \geq max(w(K_{1}), w(K_{2}))$.
Complete nonorientable minimal surfaces in a ball of $\mathbb{R}^3$
F.
J.
López;
Francisco
Martin;
Santiago
Morales
3807-3820
Abstract: The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in $\mathbb{R}^3$ with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.
The computational complexity of knot genus and spanning area
Ian
Agol;
Joel
Hass;
William
Thurston
3821-3850
Abstract: We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most $g$ is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant $C$ is NP-hard.
On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
3851-3881
Abstract: Let $X$ be a real reflexive Banach space with dual $X^{*}$ and $G\subset X$open and bounded and such that $0\in G.$ Let $T:X\supset D(T)\to 2^{X^{*}}$be maximal monotone with $0\in D(T)$ and $0\in T(0),$ and $C:X\supset D(C)\to X^{*}$ with $0\in D(C)$ and $C(0)\neq 0.$ A general and more unified eigenvalue theory is developed for the pair of operators $(T,C).$ Further conditions are given for the existence of a pair $(\lambda ,x) \in (0,\infty )\times (D(T+C)\cap \partial G)$ such that \begin{displaymath}(**)\quad\qquad\qquad\qquad\qquad\qquad\qquad Tx+\lambda Cx\owns 0.\quad\qquad\qquad\qquad\qquad\qquad\qquad\end{displaymath} The ``implicit" eigenvalue problem, with $C(\lambda ,x)$ in place of $\lambda Cx,$ is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators $T,~C.$ No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.
Calculus on the Sierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives of the heat kernel
Nitsan
Ben-Gal;
Abby
Shaw-Krauss;
Robert
S.
Strichartz;
Clint
Young
3883-3936
Abstract: This paper continues the study of fundamental properties of elementary functions on the Sierpinski gasket (SG) related to the Laplacian defined by Kigami: harmonic functions, multiharmonic functions, and eigenfunctions of the Laplacian. We describe the possible point singularities of such functions, and we use the description at certain periodic points to motivate the definition of local derivatives at these points. We study the global behavior of eigenfunctions on all generic infinite blow-ups of SG, and construct eigenfunctions that decay at infinity in certain directions. We study the asymptotic behavior of normal derivatives of Dirichlet eigenfunctions at boundary points, and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points.
Topological obstructions to certain Lie group actions on manifolds
Pisheng
Ding
3937-3967
Abstract: Given a smooth closed $S^{1}$-manifold $M$, this article studies the extent to which certain numbers of the form $\left( f^{\ast}\left( x\right) \cdot P\cdot C\right) \left[ M\right]$ are determined by the fixed-point set $M^{S^{1}}$, where $f:M\rightarrow K\left( \pi_{1}\left( M\right), 1\right)$ classifies the universal cover of $M$, $x\in H^{\ast}\left( \pi_{1}\left( M\right) ;\mathbb{Q}\right)$, $P$ is a polynomial in the Pontrjagin classes of $M$, and $C$ is in the subalgebra of $H^{\ast}\left( M;\mathbb{Q}\right)$ generated by $H^{2}\left( M;\mathbb{Q}\right)$. When $M^{S^{1}}=\varnothing$, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free $S^{1}$-action extends to an action by some semisimple compact Lie group $G$, then $\left( f^{\ast}(x)\cdot P\cdot C\right) [M]=0$. Similar vanishing results are obtained for spin manifolds admitting certain $S^{1}$-actions.
Toroidal orbifolds, gerbes and group cohomology
Alejandro
Adem;
Jianzhong
Pan
3969-3983
Abstract: In this paper we compute the integral cohomology of certain semi-direct products of the form $\mathbb{Z}^n\rtimes G$, arising from a linear $G$ action on the $n$-torus, where $G$ is a finite group. The main application is the complete calculation of torsion gerbes for six-dimensional examples arising in string theory.
An alternative approach to homotopy operations
Marcel
Bökstedt;
Iver
Ottosen
3985-3995
Abstract: We give a particular choice of the higher Eilenberg-Mac Lane maps by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial ${\bf Z}/2$-algebras.
Manifolds with an $SU(2)$-action on the tangent bundle
Roger
Bielawski
3997-4019
Abstract: We study manifolds arising as spaces of sections of complex manifolds fibering over ${\mathbb C}P^1$ with the normal bundle of each section isomorphic to $\mathcal{O}(k)\otimes {\mathbb C}^n$.
On the Cauchy problem of degenerate hyperbolic equations
Qing
Han;
Jia-Xing
Hong;
Chang-Shou
Lin
4021-4044
Abstract: In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.
Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions
Thomas
Mattman;
Katura
Miyazaki;
Kimihiko
Motegi
4045-4055
Abstract: We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The $(-3,3,5)$-pretzel knot belongs to both of the infinite families.
Scott's rigidity theorem for Seifert fibered spaces; revisited
Teruhiko
Soma
4057-4070
Abstract: We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite $\pi_1$ by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).
A moment problem and a family of integral evaluations
Jacob
S.
Christiansen;
Mourad
E. H.
Ismail
4071-4097
Abstract: We study the Al-Salam-Chihara polynomials when $q>1$. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.
Maximal theorems for the directional Hilbert transform on the plane
Michael
T.
Lacey;
Xiaochun
Li
4099-4117
Abstract: For a Schwartz function $f$ on the plane and a non-zero $v\in\mathbb{R}^2$ define the Hilbert transform of $f$ in the direction $v$ to be $\displaystyle \operatorname H_vf(x)=$p.v.$\displaystyle \int_{\mathbb{R}} f(x-vy)\; \frac{dy}y.$ Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le\vert\xi\vert\le2$, and ${\boldsymbol \zeta}f=\zeta*f$. We prove that the maximal operator $\sup_{\vert v\vert=1}\vert\operatorname H_v{\boldsymbol \zeta} f\vert$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.
The monomial ideal of a finite meet-semilattice
Jürgen
Herzog;
Takayuki
Hibi;
Xinxian
Zheng
4119-4134
Abstract: Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice, the Alexander dual is computed.
Geometric characterization of strongly normal extensions
Jerald
J.
Kovacic
4135-4157
Abstract: This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use ``group chunks'' or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.
Homomorphisms between Weyl modules for $\operatorname{SL}_3(k)$
Anton
Cox;
Alison
Parker
4159-4207
Abstract: We classify all homomorphisms between Weyl modules for $\operatorname{SL}_3(k)$ when $k$ is an algebraically closed field of characteristic at least three, and show that the $\operatorname{Hom}$-spaces are all at most one dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups.
Busemann points of infinite graphs
Corran
Webster;
Adam
Winchester
4209-4224
Abstract: We provide a geometric condition which determines whether or not every point on the metric boundary of a graph with the standard path metric is a Busemann point, that is, it is the limit point of a geodesic ray. We apply this and a related condition to investigate the structure of the metric boundary of Cayley graphs. We show that groups such as the braid group and the discrete Heisenberg group have boundary points of the Cayley graph which are not Busemann points when equipped with their usual generators.