Pseudoholomorphic discs near an elliptic point

We prove the existence and study the geometry of Bishop discs near an elliptic point of a real n-dimensional submanifold of an almost complex n-dimensional manifold.     

Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces

Let E be a generic real submanifold of an almost complex manifold. The geometry of Bishop discs attached to E is studied in terms of the Levi form of E.     

A unique continuation problem for holomorphic mappings

The authors study analytic discs that are “attached to” a red submanifold having minimal smoothness. They prove a new uniqueness and regularity theorem by using the technique of the Riemann–Hilbert problem. They also present a new method for conatructing families of analytic discs lhat osculate a surface.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

Hyperbolic embeddability of locally complete almost complex submanifolds

In this Note, we generalize to the almost complex setting, a theorem of Zaidenberg (1983) [13] and Thai (1991) [12] by giving a characterization on hyperbolic embeddability of a locally complete and relatively compact almost complex submanifold in terms of extension of pseudo-holomorphic disks from the punctured unit disk and of limit J-complex lines.     

Tangent and normal bundles in almost complex geometry

We define and study pseudoholomorphic vector bundle structures, particular cases of which are tangent and normal bundle almost complex structures. As an application we deduce normal forms of almost complex structures along a pseudoholomorphic submanifold.In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the question of non-deformation and persistence of pseudoholomorphic tori.     

J-pluripolar subsets and currents on almost complex manifolds

We prove that every almost complex submanifold of an almost complex manifold is locally J-pluripolar. This generalizes a result of Rosay for J-holomorphic submanifolds. Our second main result is an almost complex version of El Mir’s theorem for the extension of positive currents across locally complete pluripolar sets. As a consequence, we extend some results proved by Dabbek–Elkhadhra–El Mir and Dinh–Sibony in the standard complex case. We also obtain a version of the well-known results of Federer and Bassanelli for flat and \mathbb C{\mathbb {C}}-flat currents in the almost complex setting.     

Filling Real Hypersurfaces by Pseudoholomorphic Discs

We study pseudoholomorphic discs with boundaries attached to a real hypersurface E in an almost complex manifold. We give sufficient conditions for filling a one-sided neighborhood of E by the discs.      

A Normal Form for Systems of Implicit Ordinary Differential Equations

This contribution deals with the application of computer algebra based methods to the analysis of dynamic systems described by implicit ordinary differential equations. This type of system defines a submanifold of a certain jet space. We solve the problem whether this submanifold admits a parametrization by an explicit system, i.e. any solution of the original system is a solution of the explicit one and vice versa. Since this approach requires the formal integrability of the implicit system, a sketch of an algorithm for the computation of the formally integrable system is presented.     

Integrable Submanifolds in Almost Complex Manifolds

Let (M,J) be a germ of an almost complex manifold of real dimension 2m and let n (n<m) be an integer. We study a necessary and sufficient condition for M to admit an integrable submanifold N of complex dimension n. If n=m−1, we find defining functions of N explicitly from the coefficients of the torsion tensor. For J obtained by small perturbation of the standard complex structure of ℂ ^m this condition is given as an overdetermined system of second order PDEs on the perturbation. The proof is based on the rank conditions of the Nijenhuis tensor and application of the Newlander-Nirenberg theorem. We give examples of almost complex structures on ℂ³: the ones with a single complex submanifold of dimension 2 and the ones with 1-parameter or 2-parameter families of complex submanifolds of dimension 2.     

Integrability of rough almost complex structures

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

Submanifolds of generalized Hopf manifolds,type numbers and the first Chern class of the normal bundle

Summary Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.     

Defect and evaluations

Let S be a generic submanifold of C ^N of real codimension m. In this paper we continue the study, carried over by various authors, of the set of analytic discs attached to S. Moreover, we look at the subspaces of C ^N obtained by evaluating at given points, holomorphic maps which are infinitesimal deformations of analytic discs attached to S.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

Relative frames on J-holomorphic curves

The aim of this work is to prove that any non-constant J-holomorphic disc with its boundary in a given Lagrangian submanifold can be decomposed in homology into a sum of finitely many J-holomorphic simple discs with the same Lagrangian boundary condition. As a consequence, in dimension higher than 6, any generic J-holomorphic disc is multicovered.     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.     

Averaging of Legendrian Submanifolds of Contact Manifolds

We give a procedure to ‘average’ canonically C¹-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby. Mathematics Subject Classification (2000): 53D10.     

A topological minorization for the volume of vector fields on 5-manifolds

A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional.For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.Received: 29 April 2004     

Fefferman’s mapping theorem on almost complex manifolds in complex dimension two

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.Mathematics Subject Classification (2000): 32H02,53C15