Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces
Gen
Nakamura;
Yoshiaki
Maeda
1-51
Abstract: Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian $3$-manifold with nonvanishing curvature into Euclidean $6$-space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type. To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.
Cell-like mappings and nonmetrizable compacta of finite cohomological dimension
Sibe
Mardešić;
Leonard R.
Rubin
53-79
Abstract: Compact Hausdorff spaces $X$ of cohomological dimension ${\dim _Z}X \leq n$ are characterized as cell-like images of compact Hausdorff spaces $Z$ with covering dimension $Z \leq n$. The proof essentially uses the newly developed techniques of approximate inverse systems.
The space of harmonic maps of $S\sp 2$ into $S\sp 4$
Bonaventure
Loo
81-102
Abstract: Every branched superminimal surface of area $4\pi d$ in ${S^4}$ is shown to arise from a pair of meromorphic functions $ ({f_1},{f_2})$ of bidegree $ (d,d)$ such that $ {f_1}$ and ${f_2}$ have the same ramification divisor. Conditions under which branched superminimal surfaces can be generated from such pairs of functions are derived. For each $d \geq 1$ the space of harmonic maps (i.e branched superminimal immersions) of ${S^2}$ into ${S^4}$ of harmonic degree $d$ is shown to be a connected space of complex dimension $2d + 4$ .
Weakly almost periodic flows
R.
Ellis;
M.
Nerurkar
103-119
Abstract: The notion of the enveloping semigroup of a flow is applied to some situations in ergodic theory. In particular, weakly almost periodic functions on groups are studied and Moore's ergodic theorem is proved.
The spectral measure and Hilbert transform of a measure-preserving transformation
James
Campbell;
Karl
Petersen
121-129
Abstract: V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on ${L^2}$ sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.
Totally categorical structures
Ehud
Hrushovski
131-159
Abstract: A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every ${\aleph _0}$-stable, $ {\aleph _0}$-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.
Cauchy-Szeg\H o maps, invariant differential operators and some representations of ${\rm SU}(n+1,1)$
Christopher
Meaney
161-186
Abstract: Fix an integer $n > 1$. Let $ G$ be the semisimple Lie group $ {\text{SU}}(n + 1,1)$ and $ K$ be the subgroup $ {\text{S(U}}(n + 1) \times {\text{U}}(1))$. For each finite dimensional representation $ (\tau ,{\mathcal{H}_\tau })$ of $K$ there is the space of smooth $\tau $-covariant functions on $ G$, denoted by ${C^\infty }(G,\tau)$ and equipped with the action of $ G$ by right translation. Now take $ (\tau ,{\mathcal{H}_\tau })$ to be $({\tau _{p,p}},{\mathcal{H}_{p,p}})$, the representation of $K$ on the space of harmonic polynomials on $ {{\mathbf{C}}^{n + 1}}$ which are bihomogeneous of degree $(p,p)$. For a real number $\nu$ there is the corresponding spherical principal series representation of $ G$, denoted by $({\pi _\nu },{{\mathbf{I}}_{1,\nu }})$. In this paper we show that, as a $ (\mathfrak{g},K)$-module, the irreducible quotient of ${{\mathbf{I}}_{1,1 - n - 2p}}$ can be realized as the space of the $K$-finite elements of the kernel of a certain invariant first order differential operator acting on $ {C^\infty }(G,{\tau _{p,p}})$. Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of $ G$ are realized in a manner similar to Schmid's realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of $G$ with an end of complementary series representation.
Local properties of secant varieties in symmetric products. I
Mark E.
Huibregtse;
Trygve
Johnsen
187-204
Abstract: Let $L$ be a line bundle on an abstract nonsingular curve $C$, let $V \subset {H^0}(C,L)$ be a linear system, and denote by ${C^{(d)}}$ the symmetric product of $d$ copies of $C$. There exists a canonically defined $ {C^{(d)}}$-bundle map: $\displaystyle \sigma :V \otimes {\mathcal{O}_{{C^{(d)}}}} \to {E_L},$ where ${E_L}$ is a bundle of rank $ d$ obtained from $ L$ by a so-called symmetrization process. The various degenerary loci of $ \sigma$ can be considered as subsecant schemes of ${C^{(d)}}$. Our main result, Theorem 4.2, is given in $\S4$, where we obtain a local matrix description of $ \sigma$ valid (also) at points on the diagonal in ${C^{(d)}}$, and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In $\S5$ we handle the special case of giving a local scheme structure to the zero set of $\sigma$.
Local properties of secant varieties in symmetric products. II
Trygve
Johnsen
205-220
Abstract: Let $V$ be a linear system on a curve $ C$. In Part I we described a method for studying the secant varieties $ V_d^r$ locally. The varieties $V_d^r$ are contained in the $d$-fold symmetric product $ {C^{(d)}}$. In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties $V_d^1$ valid in all characteristics (Theorem 2.4). Assume $\operatorname{rk}\;V = n + 1$ and $\operatorname{char} K = 0$. In $\S\S3$ and $4$ we describe in detail the projectivized tangent cones of the varieties $V_n^1$ for a large class of points. The description is a generalization of earlier work on trisecants for a space curve. In $\S5$ we study the curve in ${C^{(2)}}$ consisting of divisors $D$ such that $ 2D \in V_4^1$ . We give multiplicity formulas for all points on this curve in $ {C^{(2)}}$ in terms of local geometrical invariants of $C$. We assume $\operatorname{char} K = 0$.
$\Delta$-closures of ideals and rings
Louis J.
Ratliff
221-247
Abstract: It is shown that if $ R$ is a commutative ring with identity and $\Delta$ is a multiplicatively closed set of finitely generated nonzero ideals of $ R$, then the operation $I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K)$ is a closure operation on the set of ideals $I$ of $R$ that satisfies a partial cancellation law, and it is a prime operation if and only if $R$ is $\Delta$-closed. Also, if none of the ideals in $ \Delta$ is contained in a minimal prime ideal, then ${I_\Delta } \subseteq {I_a}$, the integral closure of $I$ in $R$, and if $\Delta$ is the set of all such finitely generated ideals and $I$ contains an ideal in $\Delta$, then $ {I_\Delta } = {I_a}$. Further, $R$ has a natural $\Delta$-closure ${R^\Delta },A \to {A^\Delta }$ is a closure operation on a large set of rings $A$ that contain $R$ as a subring, $A \to {A^\Delta }$ behaves nicely under certain types of ring extension, and every integral extension overring of $R$ is $ {R^\Delta }$ for an appropriate set $\Delta$. Finally, if $R$ is Noetherian, then the associated primes of ${I_\Delta }$ are also associated primes of ${I_\Delta }K$ and $ {(IK)_\Delta }$ for all $K \in \Delta$.
Braids, link polynomials and a new algebra
Joan S.
Birman;
Hans
Wenzl
249-273
Abstract: A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on $2$ parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.
Integral representations of positive definite matrix-valued distributions on cylinders
Jürgen
Friedrich
275-299
Abstract: The notion of a $ G$-continuous matrix-valued positive definite distribution on $\displaystyle {S_N}(2a) \times {{\mathbf{R}}^M} \times G$ is introduced, where $ G$ is an abelian separable locally compact group and where ${S_N}(2a)$ is an open ball around zero in $ {\mathbf{R}^N}$ with radius $2a > 0$. This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable integral representation if $N = 1$ or if the matrix-valued distribution is invariant w.r.t. rotations in ${\mathbf{R}^N}$. As a consequence, appropriate extensions to the whole group are obtained. In particular, we show that a positive definite function on a certain cylinder in a separable real Hilbert space $ H$ may be extended to a characteristic function of a finite positive measure on $ H$, if it is invariant w.r.t. rotations and continuous w.r.t. a suitable topology.
Cosmicity of cometrizable spaces
Gary
Gruenhage
301-315
Abstract: A space $ X$ is cometrizable if $ X$ has a coarser metric topology such that each point of $X$ has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space $ X$ : (a) $X$ is the continuous image of a separable metric space; (b) $ {X^\omega }$ is hereditarily separable and hereditarily Lindelöf, (c) ${X^2}$ has no uncountable discrete subspaces; (d) $X$ is a Lindelöf semimetric space; (e) $ X$ has the pointed ${\text{ccc}}$. This result is a corollary to our main result which states that, under PFA, if $ X$ is a cometrizable space with no uncountable discrete subspaces, then either $X$ is the continuous image of a separable metric space or $X$ contains a copy of an uncountable subspace of the Sorgenfrey line.
On the regularity up to the boundary in the Dirichlet problem for degenerate elliptic equations
Adalberto P.
Bergamasco;
Jorge A.
Gerszonowicz;
Gerson
Petronilho
317-329
Abstract: We give a simple proof of the regularity up to the boundary of solutions of the Dirichlet problem for a class of second-order degenerate elliptic equations in the plane. We show that the method of transfer to the boundary via the associated heat equations, can be used to reduce the problem to proving the ellipticity or hypoellipticity of a pseudodifferential operator on the boundary.
An HNN-extension with cyclic associated subgroups and with unsolvable conjugacy problem
Jody Meyer
Lockhart
331-345
Abstract: In this paper, we consider the conjugacy problem for ${\text{HNN}}$-extensions of groups with solvable conjugacy problem for which the associated subgroups are cyclic. An example of such a group with unsolvable conjugacy problem is constructed. A similar construction is given for free products with amalgamation.
The lifting problem for affine structures in nilpotent Lie groups
Nguiffo B.
Boyom
347-379
Abstract: Affine manifolds occur in several situations in pure and applied mathematics, (e.g. leaves of Lagrangian foliations, completely integrable Hamiltonian systems, linear representations of virtually polycyclic groups, geometric quantization and so on). This work is devoted to left invariant affinely flat structures in Lie groups. We are mainly concerned with the following situation. Let $ G$ and ${G_0}$ be nilpotent Lie groups of dimension $n + 1$ and $n$ , respectively and let $h:G \to {G_0}$ be a continuous homomorphism from $ G$ onto ${G_0}$ . Given a left invariant affinely flat structure $ ({G_0},{\nabla _0})$ the lifting problem is to discover whether $ G$ has a left invariant affinely flat structure $ (G,\nabla)$ such that $ h$ becomes an affine morphism. In the present work we answer positively when $({G_0},{\nabla _0})$ is "normal". Therefore the existence problem for a left invariant complete affinely flat structure in nilpotent Lie groups is solved by applying the following subsequent results. Let $ \mathfrak{A}f({G_0})$ be the set of left invariant affinely flat structures in the nilpotent Lie group $ {G_0},({1^ \circ })\;\mathfrak{A}f({G_0}) \ne \emptyset $ implies the existence of normal structure $ ({G_0},{\nabla _0}) \in \mathfrak{A}f({G_0});({2^ \circ })\;h:G \to {G_0}$ being as above every normal structure $({G_0},{\nabla _0})$ has a normal lifted in $\mathfrak{A}f(G)$.
The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks
Loring W.
Tu
381-392
Abstract: A degeneracy locus is the set of points where a vector-bundle map has rank at most a given integer. Such a set is symmetric or skew-symmetric according as whether the vector-bundle map is symmetric or skew-symmetric. We prove a connectedness result, first conjectured by Fulton and Lazarsfeld, for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even ranks.
Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds
Quo-Shin
Chi;
Yunbo
Zheng
393-406
Abstract: Rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in Grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces. We prove that for such curves the curvature must be positive, and that all such simply connected curves in $ C{P^n}$ are generated by Veronese curves, thus generalizing Calabi's counterpart for holomorphic curves in $C{P^n}$. We also classify all holomorphic curves from the Riemann sphere into $G(2,4)$ whose curvature is equal to $2$ into two families, which illustrates pseudo-holomorphic curves of positive constant curvature in $G(m,N)$ are in general not unitarily equivalent, constracting to the fact that generic isometric complex submanifolds in a Kaehler manifold are congruent.
Prescribing zeros of functions in the Nevanlinna class on weakly pseudo-convex domains in ${\bf C}\sp 2$
Mei-Chi
Shaw
407-418
Abstract: Let $D$ be a bounded weakly pseudo-convex domain in $ {{\mathbf{C}}^2}$ of uniform strict type. For any positive divisor $ M$ of $D$ with finite area, there exists a holomorphic function $f$ in the Nevanlinna class such that $ M$ is the zero set of $ f$. The proof is to study the solutions of $ \bar \partial$ with ${L^1}(\partial D)$ boundary values.
On hypersurfaces of hyperbolic space infinitesimally supported by horospheres
Robert J.
Currier
419-431
Abstract: This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all eigenvalues of the second fundamental form, with respect to a particular unit normal field, be at least one. The following alternative must hold: either there is a point where all the eigenvalues of the second fundamental form are strictly greater than one, in which case the hypersurface is compact, imbedded and diffeomorphic to a sphere; or, the second fundamental form at every point has $1$ as an eigenvalue, in which case the hypersurface is isometric to Euclidean space and is imbedded in hyperbolic space as a horosphere.