arXiv research

Study of complete space-like self-expanders in Minkovski space.

Completing segments of a real tree doesn't yield a complete space.

Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.

The study extends stochastic completeness to landmark spaces with any number of landmarks.

Completeness of surface metrics established for Sobolev spaces.

Study of 3D vacuum static spaces with specific curvature properties.

A simpler proof shows $L^2$-metric completion is CAT(0).

Paper introduces a complete metric topology for low energy spaces.

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differenti…

Study on Brownian motion on discrete curve spaces, proving stochastic completeness.

Study disproves conjecture about metric completion of curve spaces.

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…

Complete Finsler spaces with negative Ricci curvature are reversible.

Lie groupoids and their orbit spaces are linked through equivalence classes.

Study on extremal subsets in geodesically complete spaces with curvature constraints.

Characterizes geodesic completeness for landmark spaces.

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves i…

The purpose of this paper is to study complete $λ$-surfaces in Euclidean space $\mathbb R^3$. A complete classification for 2-dimensional complete $λ$-surfaces in Euclidean space $\mathbb R^3$ with constant squared norm of the second fundamental form is given.

CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.

Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.

Geodesic completeness proven for certain Lorentzian spaces.

Kähler manifold loop space inherits Kähler structure and is complete.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Completes the space of vector-valued one-forms on manifolds.

Study shows how nonstandard hulls can contain metric completions.

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$ with constant squared norm of the second fundamental form.

A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactificat…

Study causal structure of warped spacetimes using novel pre-length spaces.

The study classifies complete self-shrinkers in Euclidean space.

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all…

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dy…

Study complete 3D λ-translators in Minkowski space with constant properties.

In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we c…

Negative curvature proven in Sasaki manifold space completion.

The paper describes and analyzes maximal Lorentzian surfaces with a Killing field, proving completeness criteria.

We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In th…

Researchers classify complete maximal submanifolds in pseudo-hyperbolic space.

Completed classification of Einstein spaces with specific metric properties.

We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3.

Survey on complex submanifolds in Euclidean spaces.

We complete a minor gap in Gromoll and Walschap classification of metric fibrations from the Euclidean space, thus completing the classification of Riemannian foliations on Euclidean spaces.

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

New complete minimal submanifolds found in specific Riemannian spaces.

Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.

The study of curvature in spacelike submanifolds in pseudo-hyperbolic spaces.

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for comp…