arXiv research

Totally geodesic submanifolds in hyperbolic space up to codimension two.

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

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.

In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We construct such a representative explicitly. We describe the analogous construction…

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…

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…

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

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…

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

Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.

New compact minimal submanifolds found in Riemannian symmetric spaces.

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

Generic singularities found in spacetimes with weakly trapped submanifolds.

Study finds rigid submanifolds in spacelike waves under specific conditions.

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\neq 0$ to …

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the len…

The study classifies stable submanifolds in product spaces of projective spaces.

Minimal Kaehler submanifolds up to codimension four are studied.

The paper classifies submanifolds in a specific Lorentz-Minkowski space.

Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic subma…

We show that a real Kähler submanifold in codimension $6$ is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent…

Minimal submanifolds are found as energy concentration sets in variational problems.

We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $|A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of two-convexity for hypersurface…

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.

A method constructs tractor conformal bundles for spacelike submanifolds in Lorentzian manifolds.

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…

Constructs Lorentzian manifolds from Riemannian conformal structures.

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.

Generalizes holographic method to higher codimension submanifolds.

6D symplectic manifold with many homologous but inequivalent submanifolds.

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.

We prove the vanishing of the first Chern class of a codimension 2 closed contact submanifold of a cooriented contact manifold with trivial integral 2-dimensional cohomology group. Hence the first Chern class is an obstruction for the existence of codimension 2 contact embeddings in a Darboux chart. For the existence o…

Local and global classifications of Einstein submanifolds in Euclidean space.

Exotic submanifolds in 4-manifolds remain exotic after stabilizations.

Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$ can be described by an elliptic boundary value problem, and two results about minimal…

Study on isometric submanifolds with preserved Gauss map metrics.

The paper confirms a conjecture about submanifolds in Euclidean space.

The paper proves the behavior of the second fundamental form for Kaehler submanifolds in Euclidean space.

Characterizes projective submanifolds in high dimensions.

We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of t…

Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries concerning links of codimension two. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an…

Study on heat content for submanifolds in sub-Riemannian geometry.

High codimension submanifolds evolve to convex shapes, leading to smooth limiting flows.

We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for …