Deformations and transversality of pseudo-holomorphic discs

We prove analogs of Thom’s transversality theorem and Whitney’s theorem on immersions for pseudo-holomorphic discs. We also prove that pseudoholomorphic discs form a Banach manifold.     

When is a vector field injective?

In this paper we define a notion of injectivity for a vector field over a Riemannian manifold and we give a sufficient condition for it. The proof is an extension of a well known Theorem stating that a map from an open convex set of the euclidean space into is a diffeomorphism onto its image provided that the bilinear operator associated to its differential is (positive or negative) definite everywhere. In the second part of the paper, we show that these results have a natural extension to the tangent bundle, with straightforward applications to second order differential equations. Received May 3, 1997 - Revised version received October 21, 1997     

On the injective envelope of a graded module

Abstract

Let Sing _n be the semigroup of all singular full transformations on the set X _n  = {1, 2,…, n} under the composition of functions. Let E(J _n ? 1) be the set of all idempotents of the top 𝒥-class J _n ? 1 = {α ∈ Sing _n :|im α| = n-1}. For any nonempty subset I of E(J _n  ? 1), the aim of this paper is to find a constructive necessary and sufficient condition for the semiband S(I) = ?I? to be ?-trivial. Further, the semiband S(I) is locally maximal ?-trivial if S(I) is ?-trivial and S(I ∪ {e}) is not ?-trivial for any e ∈ E(J _n ? 1 )\I. As applications, we classify locally maximal ?-trivial subsemibands and locally maximal regular ?-trivial subsemibands of Sing _n , respectively. Moreover, the characterization of which S(I) is a band is obtained.

     

Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions

We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some L^p-type estimates, standard techniques of geometric P.D.E., and some ideas from symplectic geometry and calibration theory.     

Decomposition of injective modules relative to a torsion theory

IfR is a right noetherian ring, the decomposition of an injective module, as a direct sum of uniform submodules, is well known. Also, this property characterises this kind of ring. M. L. Teply obtains this result for torsion-free injective modules. The decomposition of injective modules relative to a torsion theory has been studied by S. Mohamed, S. Singh, K. Masaike and T. Horigone. In this paper our aim is to determine those rings satisfying that every torsion-freeτ-injective module is a direct sum ofτ-uniformτ-injective submodules and also to determine those rings with the same property for everyτ-injective module.     

On the structure of pseudo-holomorphic discs with totally real boundary conditions

In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry. This paper is supported in part by NSF Grant DMS 9215011.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

On Pseudo-Holomorphic Curves from Two-Spheres into a Complex Grassmannian G（2, 5）

Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.