The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
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48 results for “Lipschitz submanifolds”
Study on transversality for Lipschitz-Fredholm maps in Fréchet spaces.
problem Existence of submanifold structure on the preimage of a transversal submanifold.
method Use of generalized Sard's theorem and transverse stability property.
result Set of Lipschitz-Fredholm maps of fixed index has transverse stability.
The paper extends a classical result to positively curved homogeneous spaces.
problem Classical results on constant functions on spheres do not extend to all positively curved homogeneous spaces.
method Proving that Lipschitz functions on positively curved homogeneous spaces are almost constant on high-dimensional submanifolds.
result Lipschitz functions on positively curved homogeneous spaces are almost constant on high-dimensional submanifolds.
Abstract relates Lipschitz-Killing measures to polar volumes of definable sets.
problem Relating geometric measures of definable sets to their polar images.
method Relates Lipschitz-Killing measures to volumes of generic polar images for smooth submanifolds, extending to infinitesimal versions.
result Establishes a relation between polar invariants and densities of generic polar images.
Paper proves compactness and finiteness of submanifolds with bounded curvature energies.
problem Establishing compactness and finiteness of submanifolds with uniformly bounded geometric curvature energies.
method Analyzes various geometric curvature energies and proves compactness and isotopy finiteness through convergence in .
result There are only finitely many isotopy types of submanifolds below a given energy value, with explicit bounds provided.
The study examines properties of projection onto differentiable manifolds.
problem Investigating the maximal domain for orthogonal projections onto submanifolds.
method Analyzes properties of the projection map on and submanifolds with Lipschitz conditions.
result Characterizes the frontier function describing the maximal domain of the projection map.
The study examines the normal growth exponent of submanifolds in negatively curved manifolds.
problem Understanding the normal growth exponent of submanifolds in negatively curved manifolds.
method Analyzing the geodesic flow and operator norms on submanifolds bi-Lipschitz to hyperbolic spaces.
result If a submanifold's normal growth exponent is at most 1, the ambient manifold is bi-Lipschitz to hyperbolic space.
Study of Anosov representations with Lipschitz limit set and applications to rigidity.
problem Characterizing Anosov representations with specific limit set properties.
method Introducing an unstable Jacobian and analyzing its orbit growth rate.
result Many higher rank representations belong to the studied class.
Hypergenerated Carnot groupsmath.MG
Characterizes hypergenerated stratified groups with flat boundaries.
problem Characterizing stratified groups with flat boundaries.
method Algebraic characterization and embedding analysis.
result Hypergenerated groups have locally bi-Lipschitz embeddings of non-characteristic hypersurfaces.
Article shows AdS-quasi-Fuchsian groups' limit sets are not smooth.
problem Smoothness of limit sets of AdS quasi-Fuchsian groups.
method Analyzes limit sets of AdS quasi-Fuchsian groups in PO(n,2).
result Limit sets are never C^1, except for Fuchsian groups.
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
Let us consider a Riemannian manifold (either separable or non-separable). We prove that, for every , every Lipschitz function can be uniformly approximated by a Lipschitz, -smooth function with $\Lip(g)\le \Lip(f)+ε$. As a consequence, every Riemannian manifold is uniformly …
Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
Study area-minimizing currents with specific boundary properties.
problem Understanding the structure of area-minimizing currents with tangentially immersed boundaries.
method Analyzes -dimensional area-minimizing currents with boundary constraints.
result Near the boundary of the current, the structure is either uncontrolled or a union of controlled hypersurfaces.
Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.
problem Investigate Möbius-invariant energies on non-smooth subsets of arbitrary dimensions.
method Show local finite energy implies embedded Lipschitz submanifold, and low fractional Sobolev regularity guarantees finite energy.
result Local graph structure of low fractional Sobolev regularity on a set is sufficient to guarantee finite energy.
New method estimates Riemannian derivatives from noisy function evaluations.
problem Optimizing functions on Riemannian manifolds with noisy data.
method Riemannian Gaussian smoothing for gradient and Hessian estimation.
result Oracle complexity independent of ambient dimension.
Properties of general Legendrian cycles acting in are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, is determined by its restriction to the Gauss curvature…
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
Geodesic connects Lagrangian graphs over torus in complex space.
problem Existence of geodesic connecting Lagrangian graphs in complex space.
method Formulated as a degenerate elliptic equation, solved via Dirichlet problem.
result Geodesic connecting Lagrangian graphs can be constructed.
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.
Curves in higher dimensions are either affine or have super-Euclidean energy growth.
problem Characterizing entire conformal curves in higher-dimensional spaces.
method Blow-down argument and interaction of generalized Cauchy--Riemann equations with calibrated geometries.
result Entire conformal curves are either affine or have super-Euclidean energy growth.
New valuations on contact manifolds generalize Euclidean integral geometry.
problem Understanding curvature in contact geometry.
method Reinterpreting valuations from Riemannian to contact manifolds.
result Contact manifolds have canonical families of generalized valuations.
The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology)…
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…
Study on vortex sheet formation in Abelian gauge theories.
problem Understanding vortex sheet formation in Abelian gauge theories.
method Inspired by Allard's regularity theory, constructs approximate solutions and analyzes their perturbations.
result Establishes a geometric framework and regularity theory for the limiting defect set.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing -Lipschitz neural networks and their density in -Lipschitz functions.
result One layer neural networks are dense in the set of all -Lipschitz functions.
New solitons found in curve metric space.
problem Elastic metric on curve spaces for shape analysis.
method Reparametrization-invariant Sobolev metric extension, geodesic equation analysis.
result Geodesics are soliton solutions for elastic metric.
Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.
problem Deformation of Lipschitz homeomorphisms
method Lipschitz analogues of Siebenmann's and Perelman's homeomorphism theory
result Lipschitz stability theorem and gluing theorem
We study Lipschitz models in reinforcement learning to bound prediction and value-function errors.
problem Bounding errors in reinforcement learning models with Lipschitz continuity constraints.
method We provide bounds on multi-step prediction error and value-function estimate using the Wasserstein metric for Lipschitz models.
result Lipschitz models lead to bounded errors in prediction and value-function estimates.
We compute the local Lipschitz constant of ReLU networks precisely.
problem Estimating the local Lipschitz constant of ReLU networks is hard.
method We use a novel approach involving the generalized Jacobian and backpropagation.
result We provide an algorithm to compute the exact Lipschitz constant of ReLU networks.
Study shows infinite distinct outer metric Lipschitz classes for knots in .
problem Lipschitz classification of surface singularities in .
method Outer metric Lipschitz equivalence and topological equivalence of knots.
result Infinitely many distinct outer metric Lipschitz classes for knots in .
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in space.
result Proves existence of differential structure from weak immersions with bounded second fundamental forms.
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
Novikov conjecture reduced to Lipschitz cohomology of groups.
problem Novikov higher signature conjecture for groups.
method Introducing Lipschitz cohomology classes and reducing the conjecture.
result Reduced Novikov conjecture to Lipschitz cohomology.
Adversarial Lipschitz Regularization improves Wasserstein GANs without gradient norm penalties.
problem Training stability and sample quality issues in Wasserstein GANs.
method Explicit Lipschitz penalty using adversarial training.
result Explicit Lipschitz penalty leads to competitive performance in Wasserstein GANs.
New MIP formulations for neural network Lipschitz constant estimation.
problem Ensuring robustness of neural networks by calculating their Lipschitz constant.
method Reformulating the neural network Lipschitz estimation problem as a Quadratically Constrained MIP (MIQCQP) problem.
result Solutions of the MIQCQP formulations provide bounds on the Lipschitz constant, with conditions for exactness.
Lipschitz-volume rigidity holds for smooth manifolds but fails for singular spaces.
problem Lipschitz-volume rigidity on singular spaces with lower curvature bounds.
method Survey of Lipschitz-volume rigidity theorems on singular spaces.
result Lipschitz-volume rigidity doesn't hold for all singular spaces.
New method for differentially private optimization with general Lipschitz conditions.
problem Differentially private optimization under general Lipschitz conditions.
method Generalized Lipschitz condition for per-sample gradients, tuning clip norm based on minimum per-sample Lipschitz constant.
result Efficacy of the recommended clip norm tuning method verified on 8 datasets.
The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.
problem Understanding the Lipschitz continuity of feature maps associated with integral kernels.
method Analyzes differentiability assumptions to derive explicit formulas for Lipschitz constants and conditions for non-Lipschitz continuity.
result Explicit formulas and conditions for Lipschitz continuity of feature maps associated with various kernels.
New scalable Lipschitz bounds improve neural network robustness analysis.
problem Computing tight Lipschitz bounds for deep neural networks is challenging and computationally expensive.
method Derived new closed-form Lipschitz bounds using more general feasible points of LipSDP, avoiding SDP solvers.
result Improved scalability and precision of Lipschitz estimation for large neural networks.
The paper studies properties of intrinsically Lipschitz constants in metric spaces.
problem Investigating properties of intrinsically Lipschitz constants.
method Introduced Leibniz and product formulas for intrinsic slope.
result Formulated Leibniz and product formulas for intrinsic slope.
New methods improve Lipschitz constant for neural network defenses.
problem Limitations of current methods for computing Lipschitz constant.
method Analyzing recent methods for computing Lipschitz constant.
result Current methods for computing Lipschitz constant have theoretical and practical limitations.
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…
Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from abovemath.DG
Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.
problem Analyzing the properties of geodesically complete spaces with curvature constraints.
method Geometric perturbations of geodesics to curves with zero length on singular sets.
result Every Sobolev map in space has a Lipschitz representative with the same Lipschitz constant as its infinity energy.
Corrects omissions in a paper about Lipschitz connectivity and invariants.
problem Lipschitz connectivity and filling invariants in solvable groups and buildings
method None specified in the abstract, focuses on correcting omissions
result Corrected omissions in a previous paper