arXiv research

Study quantifies convergence of Alexandrov spaces without collapsing.

Extends Lipschitz functions while preserving local constants.

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

Develops risk measures on Lipschitz spaces for financial positions.

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …

The paper studies properties of intrinsically Lipschitz constants in metric spaces.

New metric measure space theory for Lipschitz constants.

Proves existence of curved surfaces in hyperbolic space.

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spac…

Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.

Lipschitz and horizontal maps from an $n$-dimensional space into the $(2n+1)$-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from $S^k$ to $\H^n$ which factor through $n$-spheres and sh…

Optimizes Lipschitz estimates for partitions of unity and characterizes spaces with Assouad-Nagata dimension.

The study examines Lipschitz normally embedded Hölder triangles in 4D space.

The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.

Introduces intrinsically Lipschitz graphs in metric spaces.

The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

Let $X$ and $Y$ be length metric spaces. Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\colon X\to Y$ and $0<\mathcal H^n(X)=\mathcal H^n(f(X))<\infty$, then $f$ preserves the length of path. This property holds for …

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $…

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured la…

Proves rigidity for maps between manifolds using degree theory and current developments.

The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.

An $e^ε$-Lipschitz and co-Lipschitz map, as a metric analogue of an $ε$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stabilit…

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

We prove that any finite dimensional Alexandrov space with a lower curvature bound is locally Lipschitz contractible. As applications, we obtain a sufficient condition for solving the Plateau problem in an Alexandrov space considered by Mese and Zulkowski.

Study the landscape of Lipschitz functions between manifolds using persistent homology.

We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …

We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …

Maps between acute triangles with minimal stretch found and studied.

We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are…

Proves isometric embeddings in Euclidean spaces for RCD spaces.

This paper refines homotopy theory for cubical sets and uniform spaces.

New method improves optimization algorithms without Lipschitz smoothness.

Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.

We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston'…

Method approximates Lipschitz domains with smoother shapes.

New Transformers maintain Lipschitz continuity for robustness.

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with re…

We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

Stochastic Lipschitz bandit algorithms balance exploration and exploitation, and have been used for a variety of important task domains. In this paper, we present a framework for Lipschitz bandit methods that adaptively learns partitions of context- and arm-space. Due to this flexibility, the algorithm is able to effic…