arXiv research

Investigates intrinsic Lipschitz sections in nonlinear quotient maps.

Introduces intrinsically Lipschitz graphs in metric spaces.

Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature

This note corrects some omissions in section 2 of the paper "Lipschitz connectivity and filling invariants in solvable groups and buildings."

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

New metric measure space theory for Lipschitz constants.

The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $n$-manifold of bounded sectional curvature are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeo…

Defines new metric space sections with Ahlfors-David regularity.

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…

Paper analyzes a new Hopf-Lax semigroup in metric spaces.

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensiona…

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with …

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Defines intrinsically Hölder sections in metric spaces.

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin s…

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit…

The paper develops quantitative estimates for holomorphic sections over bounded domains.

The paper splits manifolds using infinity harmonic functions with linear growth.

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…

New bounds for low-regularity Riemannian metrics defined via distributional curvature.

In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold m…

Uniformises Kähler surfaces with positive curvature to complex plane.

The study improves harmonic map theory for metric spaces with curvature bounds.

Study limits of curved spaces with boundaries.

Langevin MCMC samples efficiently from Riemannian manifolds with geometric Euler-Murayama analysis.

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $Δ_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $δ$, absolute values of their sectional curvatures $|K_σ|\leq C$, and their injectivity radii…

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ μ_x \}_{x \in X}$ on $\partial X$, we describe a continuous family of extensions $\{\hat{f}_p : X \to Y \}_{1 \leq p \leq \infty}$ of $f$, call…

The paper explores density of stable mappings and their properties.

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 $K_{g_1} \leq -1$.…

We investigate the maximal open domain $\mathscr{E}(M)$ on which the orthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can be defined and study essential properties of $p$. We prove that if $M$ is a $C^1$ submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangent spaces, then $\ma…

Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.

Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.

The paper characterizes the geometry and topology of spin random fields.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

The paper proves a special case of Yau's conjecture for Kähler surfaces.

Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincaré condition and the measure contraction property. Using a compari…

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

Novikov conjecture reduced to Lipschitz cohomology of groups.

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arisi…

New MIP formulations for neural network Lipschitz constant estimation.

New method for differentially private optimization with general Lipschitz conditions.

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

New scalable Lipschitz bounds improve neural network robustness analysis.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…

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

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 …