arXiv research

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $δ$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second f…

Proves strong Morse inequalities for area functional in low dimensions.

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

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

Minimal Kaehler submanifolds in low codimension are often minimal.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as a…

The paper studies foliations on homogeneous spaces and identifies specific foliations.

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.

Study curve shortening flow in high dimensions with boundary constraints.

Curve shortening flow converges to a point with entropy bound.

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such …

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of $f$ by…

Extending isometric immersions with low regularity, especially supercritical.

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with cor…

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

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This statement clearly cannot be true for $C^2$ embeddings in general, due to the classi…

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…

We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal h…

The paper analyzes side effects of learning from low-dimensional data embedded in a Euclidean space.

Simple criteria for codimension two surface singularities.

Minimal submanifolds in octonionic hyperbolic spaces have large volume.

Generalizes halfspace theorems to higher dimensions for self-shrinkers.

Totally geodesic submanifolds in hyperbolic space up to codimension two.

Study wall singularities in spaces with upper curvature bounds.

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…

New examples of non-homeomorphic foliation leaves found.

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

The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.

The paper classifies and studies conformal variations of submanifolds.

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

The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…

We construct and embedding of a Nöbeling space $N^n_{n-2}$ of codimension $2$ into a Menger space $M^n_{n-2}$ of codimension $2$. This solves an open problem stated by R.~Engelking in 1978 in codimension~$2$.

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exh…

This paper classifies embedded, codimension-one spheres which are null homotopic. This information is used to show that all null homotopic, immersed codimension-one spheres which are taut in the sense of Terng and Thorbergsson are actually distance spheres.

Study how pairs of 1D foliations can be deformed into contact structures.

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb H P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is ev…

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 of codimension-1 embeddings in 3-manifolds using twist maps and push maps.

This paper extends NCFI to odd codimension and computes examples.

Study shows singular set of certain graphs has codimension 1.

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with $|A|^2\le 1$ in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.