arXiv research

The study provides optimal estimates for surfaces close to constant mean curvature.

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

Existence and rigidity results for lifts in Carnot groups.

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…

Bi-Lipschitz Autoencoder ensures robust manifold preservation.

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover, if $M$ has finite volume, the equality holds if and only if the sectional curvat…

The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.

Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…

The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.

Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.

This is the first paper of two ones. Here we prove that two compact Alexandrov surfaces of bounded integral curvature having no peak points are bi-Lipschitz equivalent if they are homeomorphic one to the other. Also conditions under that two ends having finite integral negative curvature are bi-Lipschitz equivalent are…

Bi-Lipschitz flows approximate a wide range of distributions.

Let $X$ be a closed semialgebraic set of dimension $k.$ If $n\ge 2k+1$, then there is a bi-Lipschitz and semialgebraic embedding of $X$ into $\Bbb R^n.$ Moreover, if $n \ge 2k+2$, then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of $\Bbb R^n.$

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

Researchers develop a new framework to control neural network sensitivity.

Study links between surface germs and knot theory in 4D.

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0,…

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R…

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

In this paper we describe the notion of a weak lipschitzianity of a mapping on a $C^{q}$ stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the …

Proves isometric embeddings in Euclidean spaces for RCD spaces.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

Characterizes hypergenerated stratified groups with flat boundaries.

In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of …

In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with $C^{1,α}$, $0<α\le 1$, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two …

The paper sharpens a theorem about surfaces with zero Gaussian curvature.

We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty, exploited in Gromov's proof of the systolic inequality for essential manifolds, admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every C>0. Our key tool is the first variation formula thought of…

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manif…

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lip…

This is a continuation of the joint paper with the same title by A.Belenkiy and Yu.Burago. It is proved here that two homeomorphic closed Alexandrov surfaces (of bounded integral curvature) are bi-Lipschitz with a constant depending only on upper bounds of their Euler number, diameters, negative integral curvatures, an…

Study Ricci-Deturck flow from rough metrics, proving short-time existence.

We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also apply for bounded subsets of complete Riemannian manifolds, and complete flat a…

Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a compact in $X$, and with sectional curvatures $K_{g_1}$ satisfying $K_{g_1} \leq -1$.…

The study connects polyhedral manifolds to Riemannian ones with geometric bounds.

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

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. Th…

Study improves Poincaré-Sobolev inequalities for differential forms.

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.

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the th…

We study here limit spaces $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with c…

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…