Submanifolds Of Codimension Research Articles

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

Complete real Kähler submanifolds in codimension two

Minimal isometric immersions $$f : M^{2n} \rightarrow {\mathbb{R}}^{2n+2}$$ in codimension two from a complete Kähler manifold into Euclidean space had been classified in Dajczer and Gromoll (Invent Math 119:235–242, 1995) for n ≥ 3. In this note we describe the non-minimal situation showing that, if f is real analytic but not everywhere minimal, then f is a cylinder over a real Kähler surface $$g : N^4 \rightarrow {\mathbb{R}}^6$$ , that is, $$M^{2n} = N^4 \times {\mathbb{C}}^{n-2}$$ and f = g × id split, where $${id} : {\mathbb{C}}^{n-2} \cong {\mathbb{R}}^{2n-4}$$ is the identity map. Moreover, g can be further described.

Some Power Series Expansions for Monogenic Functions

New series developments for monogenic functions are presented. The terms of these series have factors that are expressible as power functions vanishing on special higher codimension submanifolds of Euclidean space. These series are closely related with the Cauchy-Kowalewski extension problem as well as to special Vekua systems arising from the consideration of axial and biaxial symmetry.

Gromov-Witten invariants and pseudo symplectic capacities

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.

Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Δ u=up in Ω, u=0 on ∂Ω, where Ω is a bounded domain of \(\mathcal{R}^n\), n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and \(p\ge {2n\over n-2}\), as a consequence of the Pohožaev's identity).

Geodesics, the equivalence principle and singularities in higher-dimensional generalrelativity and braneworlds

The geodesics of a spacetime seldom coincide with those of an embedded submanifold ofco-dimension one. We investigate this issue for higher-dimensional general relativity-likemodels, firstly in the simpler case without branes to isolate which features are alreadypresent, and then in the more complicated case with branes. The framework in which weconsider branes is general enough to include asymmetric braneworlds but not thick branes.We apply our results on geodesics to study both the equivalence principle andcosmological singularities. Among the models that we study, these considerations favourZ2 symmetric braneworlds with a negative bulk cosmological constant.

SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 OF A COMPLEX PROJECTIVE SPACE IN TERMS OF THE JACOBI OPERATOR

In this paper, we characterize some semi-invariant sub-manifolds of codimension 3 with almost contact metric structure (<TEX>$\phi$</TEX>, <TEX>$\xi$</TEX>, g) in a complex projective space <TEX>$CP^{n+1}$</TEX> in terms of the structure tensor <TEX>$\phi$</TEX>, the Ricci tensor S and the Jacobi operator <TEX>$R_\xi$</TEX> with respect to the structure vector <TEX>$\xi$</TEX>.

On the horospherical ridges of submanifolds of codimension 2 in Hyperbolic n-space

We introduce the notion of horospherical ridges for submanifolds of codimension 2 in hyperbolic n-space, and study some of their properties.

Genuine Rigidity of Euclidean Submanifolds in Codimension Two

An isometric deformation of an Euclidean submanifold is called genuine if the submanifold cannot be included into a submanifold of larger dimension in such a way that the deformation of the former is given by an isometric deformation of the latter. The submanifold is said to be genuinely rigid if it has no genuine deformations. In this paper we study the deformation problem in codimension two for the classes of elliptic and parabolic submanifolds. In spite of having a second fundamental form as degenerate as possible without being flat, i.e., the Gauss map has rank two everywhere, our main result says that generically these submanifolds are genuinely rigid. An additional unexpected deformation phenomenon for elliptic submanifolds carrying a Kaehler structure is described.

Schwarz-type lemmas for solutions of $\bar\partial $-inequalities and complete hyperbolicity of almost complex manifolds

The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.

NORMAL SYSTEMS OF COORDINATES ON MANIFOLDS OF CHERN-MOSER TYPE

It is known that the CR geometries of Levi non-degen-erate hypersurfaces in <TEX>$\C^n$</TEX> and of the elliptic or hyperbolic CR submanifolds of codimension two in <TEX>$\C^4$</TEX> share many common features. In this paper, a special class of normalized coordinates is introduced for any CR manifold M which is one of the above three kinds and it is shown that the explicit expression in these coordinates of an isotropy automorphism <TEX>$f{\in}Aut(M)_o {\subset}Aut(M),\;o{\in}M$</TEX>, is equal to the expression of a corresponding element of the automorphism group of the homogeneous model. As an application of this property, an extension theorem for CR maps is obtained.

ON SOME SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX PROJECTIVE SPACE

In this paper, We characterize a semi-invariant sub-manifold of codimension 3 satisfying ∇<TEX>$\varepsilon$</TEX>A = 0 in a complex projective space CP<TEX>$\^$</TEX>n+1/, where ∇<TEX>$\varepsilon$</TEX>A is the covariant derivative of the shape operator A in the direction of the distinguished normal with respect to the structure vector field <TEX>$\varepsilon$</TEX>.

ON A SEMI-INVARIANT SUBMANIFOLD OF CODIMENSION 3 WITH CONSTANT MEAN CURVATURE IN A COMPLEX PROJECTIVE SPACE

Let M be 3 Semi-invariant submanifold of codimension 3 with lift-flat normal connection in a complex projective space. Further, if the mean curvature of M is constant, then we prove that M is a real hypersurface of a complex projective space of codimension 2 in the ambient space.

Long-time asymptotics of the Navier-Stokes and vorticity equations on R(3).

We use the vorticity formulation to study the long-time behaviour of solutions to the Navier-Stokes equation on R(3). We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field satisfies ||u(.,t)||(L(2)) = o(t(-5/4)) as t-->+ infinity. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.

On the flat ridges of submanifolds of codimension 2 in Rn

We introduce the concept of flat ridges for submanifolds of codimension 2 from the viewpoint of their contact with hyperplanes. We characterize them geometrically, studying some of their properties. In particular, we see that the highest-order flat ridges coincide with the flattenings of the asymptotic lines and from this we obtain some lower bounds for their numbers under appropriate conditions.

On the flat ridges of submanifolds of codimension 2 in ℝ

On the flat ridges of submanifolds of codimension 2 in ℝ

J-INVARIANT SUBMANIFOLDS OF CODIMENSION 2 IN A COMPLEX PROJECTIVE SPACE

In this paper we prove that if M is a J-invariant sub-manifold of codimension 2 in a complex projective space <TEX>$P_{n+1}(C)$</TEX>, and the second fundamental tensor is cyclic-parallel or M has harmonic curvature, then M is locally complex quadric Q<TEX>$_n$</TEX>(C) or P<TEX>$_n$</TEX>(C).

Maximum Modulus Sets and Segre Convexity

Let E be a totally real, analytic, n-dimensional manifold, foliated by analytic interpolation submanifolds of codimension 1, in the analytic boundary of a Segre-convex domain in ℂn. Given a canonical defining function of the boundary of Ω in a point 0 of E : Im z1 +R[Re z1, z′, z′]=0. If all the odd exponents in the decomposition of R in irreducible factors, at 0,are greater than 1 then R≥0 and E is locally contained in a maximum modulus set.

Splitting Obstruction Groups in Codimension 2

The splitting obstruction groups depend functorially on the square of fundamental groups. In the paper the problem of splitting along a submanifold of codimension two under some restrictions on the square of fundamental groups is considered. New exact sequences and commutative diagrams containing Wall groups, splitting obstruction groups, and surgery obstruction groups for manifold pairs are obtained. Examples of computation of splitting obstruction groups and natural maps are considered.