We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

In this article we show that every smooth closed oriented fourmanifold admits a decomposition into two submanifolds along common boundary. Each of these submanifolds is a complex manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-manifold is a Stein domain in the the complement of a certain contractible 2-complex.

We study the differential geometric properties of n-dimensional real submanifolds M of complex space forms M , whose maximal holomorphic tangent subspace is (n− 1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space. Using certain condition on the induced almost contact structure F and on the second fundamental form h of t...

A local uniqueness property of holomorphic functions on realanalytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic sub...

Given the Lagrangian fibration T 4 → T 2 and a Lagrangian submanifold, exhibiting an elliptic umbilic and supporting a flat line bundle, we study, in the context of mirror symmetry, the “quantum” corrections necessary to solve the monodromy of the holomorphic structure of the mirror bundle on the dual fibration. 2000 Mathematics Subject Classification: 14J32, 37G25, 51P05, 53D12, 70K60, 81T30 E...

Abstract We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral $$\Psi ^{(H)}(x,y,z):= \int _{-\infty }^{\infty }\int }\exp (i(s^3+t^3+zst$$ Ψ ( H ) x , y</m...

In this paper, we investigate properties of orbits Hermann actions as submanifolds without assuming the commutability involutions which define actions. particular, compute second fundamental form actions, and give a sufficient condition for to be weakly reflective arid submanifolds.

In this paper, we study warped products of contact skew-CR submanifolds, called skew CR-warped products. We establish a lower bound relationship between the squared norm second fundamental form and warping function. The equality case inequality is investigated some special cases derived are given. Furthermore, provide non-trivial examples such submanifolds.

We discuss the behaviour of vertices and inflexions of one-parameter families of plane curves which include a singular member. These arise a sections of smooth surfaces by families of planes parallel to the tangent plane at a given point. We cover all the generic cases, namely elliptic (A1), umbilic (A1), hyperbolic (A1), parabolic (A2) and cusp of Gauss (A3) points. This work is preliminary to...

Novel caustic phenomena, which contain fold, butterfly and elliptic umbilic catastrophes, are observed in defocused images of two approximately collinear oppositely biased metallic tips in a transmission electron microscope. The observed patterns depend sensitively on defocus, on the applied voltage between the tips and on their separation and lateral offset. Their main features are interpreted...