arXiv research

Research shows how certain flat structures behave in specific convex domains.

Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.

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

Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.

Exotic submanifolds in 4-manifolds remain exotic after stabilizations.

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

Classifies hyperbolic groups with surface-like boundaries.

We study relatively hyperbolic Coxeter groups of type $HM$ with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.

Study shows singular set of certain graphs has codimension 1.

In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.

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

New examples of isoparametric foliations on $S^n imes S^n$ are provided.

Study critical points of Laplace eigenfunctions in polygons.

We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the…

Finite element method approximates scalar curvature in arbitrary dimensions.

We study local ($n+1$)-webs of codimension 1 on a manifold of dimension $n.$ We give a complete description of their possible Lie algebras of infinitesimal diffeomorphisms. More precisely we show that these Lie algebras are direct products of sub-algebras which are isomorphic to $\frak{sl}(2),$ to the non-commutative 2…

Stokes' theorem's boundary maximizes entropy.

Study shows how compact shapes can be rigidly mapped into complete manifolds.

On compact Kähler manifolds, we classify regular holomorphic foliations of codimension 1 whose canonical bundle is numerically trivial.

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 simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyper…

Alternative approach to rigidity of high-dimensional isometric immersions.

Many compactly generated pseudo-groups of local transformations on 1-manifolds are realizable as the transverse dynamic of a foliation of codimension 1 on a compact manifold of dimension 3 or 4.

Manifolds can be dominated by hypersurfaces in a sphere.

The purpose of this paper is to give an effective construction for some induced structures on spheres or product of spheres of codimension 1, 2 or 3, respectively, in Euclidean space endowed with an almost product structure.

The paper studies affine manifolds with linear foliations and their topological properties.

The paper develops formulas for hyperbolic simplices based on edge lengths.

Generalizes halfspace theorems to higher dimensions for self-shrinkers.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.

We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit such a foliation.

The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an…

We construct a groupoid equivariant Kasparov class for transversely oriented foliations in all codimensions. In codimension 1 we show that the Chern character of an associated semifinite spectral triple recovers the Connes-Moscovici cyclic cocycle for the Godbillon-Vey secondary characteristic class.

The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.

In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.

We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^…

On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.

Study PL bordism theories with quantitative bounds on filling simplices.

Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 Bäcklund transformations relating two hyperbolic Monge-Ampère systems. In cases when the Bäcklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new e…

Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.

New framework shows $C^*$-simplicity for groups without certain subalgebras.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.