arXiv research

Minimal isometric immersions F in codimension two from a complete Kahler manifold into Euclidean space had been classified for dimension greater than or equal to 3. In this note we describe the non--minimal situation by showing that, if F is real analytic but not everywhere minimal, then F is a cylinder over a real Kah…

The paper flattens a non-degenerate CR singular point in complex space.

Minimal real Kähler submanifolds in codimension 6 are holomorphic.

In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states th…

Study complex slices on real algebraic varieties and their properties.

The paper characterizes stable cohomotopy groups in codimensions two and three, linking algebraic and geometric perspectives.

Simple criteria for codimension two surface singularities.

Totally geodesic submanifolds in hyperbolic space up to codimension two.

In this paper we study submanifolds of almost contact manifolds with Norden metric of codimension two with totally real normal spaces. Examples of such submanifolds as a Lie subgroups are constructed.

In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimension…

Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.

Research explores real algebraic realization of round fold maps of codimension -1.

This paper classifies Kaehler submanifolds in hyperbolic space with low codimension.

Minimal Kaehler submanifolds up to codimension four are studied.

Study wall singularities in spaces with upper curvature bounds.

Study shows codimension 2 foliations on simply-connected manifolds are smooth.

Lu's conjecture proven for minimal surfaces in codimension two.

Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.

Classifies polar foliations on symmetric spaces.

A local uniqueness property of holomorphic functions on real-analytic 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 stu…

After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for non degenerate codimensio…

In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.

Researchers found counterexamples to a 2-jet determination theorem in higher codimension.

Study proves equality of signature classes in codimension two.

Study shows non-abelian free groups' 4th cohomology is non-zero.

Study on fractional mass for codimension-two currents, proving equi-coercivity and Γ-convergence.

We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null accel…

Study on Hermitian metrics on Lie algebras with specific ideals.

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by …

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umb…

Study extends knot width concept to higher dimensions, finding large widths possible.

Study on flat singularities of area-minimizing currents in codimension one.

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…

For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually…

We show that among the Euclidean submanifolds with codimension two the ones of rank two that are parabolic but nonruled are isometrically rigid. This generalizes the result in [10] that these submanifolds are genuinely rigid. In addition, we give a parametric classifications of all parabolic submanifolds.

Two codimension-one submanifolds are cobordant if they have the same homology class.

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebra…

The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also det…

The paper analyzes high codimension mean curvature flow and proves cylindrical estimates near singularities.

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. Th…

Willmore surfaces have umbilic points forming a smooth manifold.

In this paper we prove that a complete noncompact manifold with nonnegative Ricci curvature has a trivial codimension one homology unless it is a split or flat normal bundle over a compact totally geodesic submanifold. In particular, we prove the conjecture that a complete noncompact manifold with positive Ricci curvat…

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…

Connected sums defined for codimension two locally flat submanifolds in higher dimensions.

On a 6-dimensional real vector space $V$ there are three types of multisymplectic 3-forms. We present in this paper a unified treatment of these three types. Forms of each type represent a subset of $Λ^3 V^*$. In two cases they are open subsets, in the third one it is a submanifold of codimension 1. We study the geomet…

Proves codimension of abnormal set in step 2 Carnot groups is at least 3.

In this paper, we study a complete noncompact nonnegatively curved Alexandrov space $A$ with a soul $S$ of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, $π:\ A\to S$, is a submetry.

The paper shows conditions under which a Kaehler submanifold is minimal or holomorphic.