arXiv research

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…

Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.

In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obt…

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

This paper characterizes which subsets of C^n can be the set of positions of n points on a linkage in the complex plane C. For example, assuming compactness they are just compact semialgebraic sets. Noncompact configuration spaces are semialgebraics sets invariant under the Euclidean group, with compact quotient.

We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2…

We consider a complete biharmonic immersed submanifold $M$ in an Euclidean space $\mathbb{E}^N$. Assume that the immersion is proper, that is, the preimage of every compact set in $\mathbb{E}^N$ is also compact in $M$. Then, we prove that $M$ is minimal. It is considered as an affirmative answer to the global version o…

Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

Paper presents a new Pohozaev-Schoen identity for non-compact manifolds.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

The study explores dilating set properties across Euclidean and hyperbolic geometries.

We consider a complete nonnegative biminimal submanifold M (that is, a complete biminimal submanifold with lambda>=0) in a Euclidean space E^N. Assume that the immersion is proper, that is, the preimage of every compact set in E^N is also compact in M. Then, we prove that M is minimal. From this result, we give an affi…

Compact gradient ρ-Einstein solitons are isometric to Euclidean spheres.

The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.

For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors h…

Magnitude of Euclidean domains predicts Willmore energy in odd dimensions.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

In this paper, we construct compact embedded $λ$-hypersurfaces with the topology of torus which are called $λ$-torus in Euclidean spaces $\mathbb {R}^{n+1}$.

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

Logarithmic Sobolev inequality proven for non-compact self-shrinkers.

We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…

The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.

New examples of isoparametric families on non-compact symmetric spaces.

Paper links set derivatives to its orthogonal projections.

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of the Euclidean three dimensional space. In a recent preprint J. Jorge and F. Merc…

In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either $\frac{n^2}{4}$ or $\frac{(n-2)^2}{4|x|^2}$. Mor…

A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Proves isometric embeddings in Euclidean spaces for RCD spaces.

In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spac…

I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space. The main result roughly states that the class of these spaces coincides with class of inverse limits of Euclidean d-polyhedra.

Curvature conditions distinguish Euclidean space and disks in contractible manifolds.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.

New stability and isolation results for Einstein manifolds.

We study global injectivity of proper branched coverings defined on the Euclidean $n$-ball in the case when the branch set is compact. In particular we show that such mappings are homeomorphisms when $n=3$ or when the branch set is empty. This proves the corresponding cases of a question of Vuorinen from [Vuo79].

Theory proves existence of hypersurfaces with prescribed curvature.

We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties …

Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion…

The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption i…

Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.

Photo-identification (photo-id) of dolphin individuals is a commonly used technique in ecological sciences to monitor state and health of individuals, as well as to study the social structure and distribution of a population. Traditional photo-id involves a laborious manual process of matching each dolphin fin photogra…

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starsh…