Subspaces of non-commutative spaces
S.
Paul
Smith
2131-2171
Abstract: This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space.
Cotensor products of modules
L.
Abrams;
C.
Weibel
2173-2185
Abstract: Let $C$ be a coalgebra over a field $k$ and $A$ its dual algebra. The category of $C$-comodules is equivalent to a category of $A$-modules. We use this to interpret the cotensor product $M \square N$ of two comodules in terms of the appropriate Hochschild cohomology of the $A$-bimodule $M \otimes N$, when $A$ is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.
Local subgroups and the stable category
Wayne
W.
Wheeler
2187-2205
Abstract: If $G$ is a finite group and $k$ is an algebraically closed field of characteristic $p>0$, then this paper uses the local subgroup structure of $G$to define a category $\mathfrak{L}(G,k)$ that is equivalent to the stable category of all left $kG$-modules modulo projectives. A subcategory of $\mathfrak{L}(G,k)$ equivalent to the stable category of finitely generated $kG$-modules is also identified. The definition of $\mathfrak{L}(G,k)$ depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of $p$-local subgroups.
Partial regularity for the stochastic Navier-Stokes equations
Franco
Flandoli;
Marco
Romito
2207-2241
Abstract: The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time $t$ the set of singular points is empty. The same result holds true for every martingale solution starting from $\mu_0$-a.e. initial condition $u_0$, where $\mu_0$ is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure $\mu_0$ is supported on the whole space $H$ of initial conditions.
Random points on the boundary of smooth convex bodies
Matthias
Reitzner
2243-2278
Abstract: The convex hull of $n$ independent random points chosen on the boundary of a convex body $K \subset \mathbb{R}^d$ according to a given density function is a random polytope. The expectation of its $i$-th intrinsic volume for $i=1, \dots, d$ is investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as $n \to \infty$ are derived.
Generalized space forms
Neil
N.
Katz;
Kei
Kondo
2279-2284
Abstract: Spaces with radially symmetric curvature at base point $p$ are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to ${\mathbb R^n}$ or $S^n$ under a radially symmetric metric, to ${\mathbb R}{\rm P}^n$ with Riemannian universal covering of $S^n$equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at $p$.
A momentum construction for circle-invariant Kähler metrics
Andrew
D.
Hwang;
Michael
A.
Singer
2285-2325
Abstract: Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea--the ``Calabi ansatz''--is to begin with an Hermitian line bundle $p:(L,h)\to(M,g_M)$ over a Kähler manifold, and to search for Kähler forms $\omega=p^*\omega_M+dd^c f(t)$ in some disk subbundle, where $t$ is the logarithm of the norm function and $f$ is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric $g$ arising from the Calabi ansatz. This suggests geometric hypotheses (which we call ``$\sigma$-constancy'') to impose upon the base metric $g_M$ and Hermitian structure $h$ in order that the scalar curvature of $g$ be specified by solving an ODE. We show that $\sigma$-constancy is ``necessary and sufficient for the Calabi ansatz to work'' in the following sense. Under the assumption of $\sigma$-constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to $g_M$ on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of $\sigma$-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on $\mathbf{C}^2$.
Isoperimetric regions in cones
Frank
Morgan;
Manuel
Ritoré
2327-2339
Abstract: We consider cones $C = 0\, \times{\kern-10.5pt}\times \,M^n$ and prove that if the Ricci curvature of $C$ is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.
Orthogonal polynomials and quadratic extremal problems
J.
M.
McDougall
2341-2357
Abstract: The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.
Three-divisible families of skew lines on a smooth projective quintic
Slawomir
Rams
2359-2367
Abstract: We give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3. We study properties of the codes given by arrangements of disjoint lines on quintics.
Twisted face-pairing 3-manifolds
J.
W.
Cannon;
W.
J.
Floyd;
W.
R.
Parry
2369-2397
Abstract: This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing $\epsilon$ of a faceted 3-ball $P$ yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold $P/\epsilon$ systematically. The method describes how to modify $P$ by edge subdivision and how to modify any orientation-reversing face-pairing $\epsilon$ of $P$ by twisting, so as to yield an infinite parametrized family of face-pairings $(Q,\delta)$ whose quotient complexes $Q/\delta$ are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings $\epsilon$, it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex $Q/\delta$ of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to $Q$. (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings. We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds $Q/\delta$ and of the original pseudomanifold $P/\epsilon$ (with vertices removed). We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.
The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition
Antoine
Henrot;
Henrik
Shahgholian
2399-2416
Abstract: Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the $p$-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure $a(x)$ on the ``free'' streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function $a(x)$ is subject to certain convexity properties. In our earlier results we have considered the case of constant $a(x)$. In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the $p$-capacitary potentials in convex rings, with $C^1$ boundaries.
Ergodic and Bernoulli properties of analytic maps of complex projective space
Lorelei
Koss
2417-2459
Abstract: We examine the measurable ergodic theory of analytic maps $F$ of complex projective space. We focus on two different classes of maps, Ueda maps of ${\mathbb P}^{n}$, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy ( $h_{\mu}(F) = h_{top}(F)= \log(\deg F)$). We find analytic maps of ${\mathbb P}^1$ and ${\mathbb P}^2$ which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer $d>1$, there exists a rational map of the sphere which is one-sided Bernoulli of entropy $\log d$ with respect to a smooth measure.
Hyponormality of trigonometric Toeplitz operators
In
Sung
Hwang;
Woo
Young
Lee
2461-2474
Abstract: In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators $T_{\varphi }$ with trigonometric polynomial symbols $\varphi$. Our criterion involves the zeros of an analytic polynomial $f$ induced by the Fourier coefficients of $\varphi$. Moreover the rank of the selfcommutator of $T_{\varphi }$ is computed from the number of zeros of $f$ in the open unit disk $\mathbb{D}$ and in $\mathbb{C}\setminus \overline{\mathbb{D}}$ counting multiplicity.
Topological dynamics on moduli spaces II
Joseph
P.
Previte;
Eugene
Z.
Xia
2475-2494
Abstract: Let $M$ be an orientable genus $g>0$ surface with boundary $\partial M$. Let $\Gamma$ be the mapping class group of $M$ fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \operatorname{Hom}_{\mathcal C}(\pi_1(M),\operatorname{SU}(2))/\operatorname{SU}(2),$ the space of $\operatorname{SU}(2)$-gauge equivalence classes of flat $\operatorname{SU}(2)$-connections on $M$ with fixed holonomy on $\partial M$. We study the topological dynamics of the $\Gamma$-action and give conditions for the individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}$.
Algebraic and spectral properties of dual Toeplitz operators
Karel
Stroethoff;
Dechao
Zheng
2495-2520
Abstract: Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.
Regularized orbital integrals for representations of ${\mathbf{S} \mathbf{L}}(2)$
Jason
Levy
2521-2539
Abstract: Given a finite-dimensional representation of ${\mathbf{S} \mathbf{L}}(2,F)$, on a vector space $V$ defined over a local field $F$ of characteristic zero, we produce a regularization of orbital integrals and determine when the resulting distribution is non-trivial.
Some convolution inequalities and their applications
Daniel
M.
Oberlin
2541-2556
Abstract: We introduce a class of convolution inequalities and study the implications of these inequalities for certain problems in harmonic analysis.