arXiv research

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.

Proves isometric embeddings in Euclidean spaces for RCD spaces.

Paper proves total curvature for convex hypersurfaces in equiaffine space.

Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a sub…

Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.

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

Study 2D spaces with curvature, finding a graph structure.

Heat kernels map RCD spaces to Riemannian manifolds.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

Let $Σ$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:Σ\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differen…

Gradient flow of curve length on Sobolev metrics preserves convexity.

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is …

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…

The paper confirms a conjecture about manifolds with positive curvature.

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature mea…

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theo…

In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the p…

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has…

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statemen…

Alternative proof and extension of curvature estimates for minimal immersions.

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (includin…

The paper proves the existence of hypersurfaces with prescribed mean curvature.

The paper proves existence of minimal homotopies for immersed planar curves.

We show that a map with Hölder exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov fo…

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of…

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. …

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.

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

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

We compute the local Lipschitz constant of ReLU networks precisely.

Lawson-Osserman constructed three types of non-parametric minimal cones of high codimensions based on Hopf maps between spheres, which correspond to Lipschitz but non-differentiable solutions to the minimal surface equations, thereby making sharp contrast to the regularity theorem for minimal graphs of codimension 1. I…

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.

The study constructs immersions with controlled curvatures between manifolds and identifies obstacles.

New method for differentially private optimization with general Lipschitz conditions.

Proves bijection between smooth conformal immersions and immersions.

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…

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…

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 …