arXiv research

Characterizes geodesic completeness for landmark spaces.

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

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

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

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

Kähler manifold loop space inherits Kähler structure and is complete.

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

Geodesic completeness proven for certain Lorentzian spaces.

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

The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.

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.

Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.

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

I show that if a geodesic space has curvature bounded below locally in the sense of Alexandrov then its completion has the same lower curvature bound globally.

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

Completeness of surface metrics established for Sobolev spaces.

Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.

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

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…

New metric spaces for geodesic rays in cohomology classes.

Geodesics found in a metric space of m-subharmonic functions.

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

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

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the…

Study on completeness of Sobolev metrics on manifold-valued curves.

It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.

Geodesic completeness proven for certain symmetric spaces.

The study provides bounds for geodesic diameter in Euclidean space.

We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-c…

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half…

We create a smooth manifold of triangular meshes with a geodesically complete metric.

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

Study classifies polyharmonic helices in various space forms.

Totally geodesic dual leaves on curved manifolds are also curved.

A new Riemannian metric on curve spaces is complete and smooth.

We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize len…

Method constructs WP geodesics to study Teichmüller space behavior.

Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichmüller space. We sketch a collection of results by other authors and then proceed to develop the properties of the WP CAT(0) geometry. Our approach includes a simplified proof of the Masur-Wolf …

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also con…

Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was given in \cite{drt} for Euclidean hypers…

This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length $\le t$; then $\lim\limits_{t \to \infty} P…

Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise s…

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spac…