Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.
problem Classifying semi-algebraic surfaces with isolated singularities.
method Bi-Lipschitz homeomorphisms with inner distance.
result Complete classifications for Nash surfaces and complex algebraic curves.
Study shows infinite distinct outer metric Lipschitz classes for knots in S3.
problem Lipschitz classification of surface singularities in R4. method Outer metric Lipschitz equivalence and topological equivalence of knots.
result Infinitely many distinct outer metric Lipschitz classes for knots in S3. Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in R4. method Constructing surface germs XK linked to knots K in S3 and studying their Lipschitz geometry. result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
problem Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
method Normalized Ricci flow on compact surfaces.
result Uniform Lipschitz continuity of isoperimetric profiles under normalized Ricci flow.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
The study proves surfaces in a specific Heisenberg group must be simple planes.
problem Characterizing surfaces in a sub-Finsler Heisenberg group.
method Analyzes (X,Y)-Lipschitz surfaces in H1 with a sub-Finsler structure. result Complete, oriented, stable (X,Y)-Lipschitz surfaces are vertical planes. Lipschitz maps on metric surfaces are rigid if they preserve area.
problem Understanding the rigidity of Lipschitz maps on metric surfaces.
method Established a coarea inequality for continuous Sobolev functions on metric surfaces.
result Proved that 1-Lipschitz maps from a closed metric surface to a closed Riemannian surface preserving area are isometries.
The study provides optimal estimates for surfaces close to constant mean curvature.
problem Optimizing estimates for surfaces near constant mean curvature.
method Bi-Lipschitz and W2,2 parametrization for surfaces with density close to one and small Willmore energy. result Quantitative rigidity for L2-almost CMC surfaces. 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 …
Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.
problem Lipschitz homology classes and their moduli.
method Defining a new modulus dModp and proving a homological duality theorem. result Every relative Lipschitz k-homology class has a unique dual class satisfying a specific modulus product equality. The paper examines partial regularity of Lipschitz solutions to minimal surface system.
problem Understanding the regularity of solutions to the minimal surface system.
method Investigation of stationary, integral weak, and viscosity solutions; interior gradient estimate using maximum principle.
result Partial regularity results for Lipschitz solutions, including interior gradient estimate.
We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.
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…
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
Study Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
problem Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
method Generalization of ROF model, existence and uniqueness of minimizers, regularity results on PDE system.
result Lipschitz regularity of minimizers without convexity requirements.
Lipschitz mappings found between Riemann surfaces with specific properties.
problem Finding globally Lipschitz mappings between doubly connected Riemann surfaces.
method Using a result from Iwaniec, Kovalev, and Onninen, the minimizer of the energy functional is shown to be locally Lipschitz and globally Lipschitz.
result The minimizer of the energy functional is a globally Lipschitz mapping.
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…
Paper defines and studies measures related to earthquakes and best Lipschitz maps.
problem Understanding Thurston's conjecture about maps and measures on hyperbolic surfaces.
method Examining Lie algebra valued transverse measures and their relation to earthquakes.
result Defines and shows correspondence between best Lipschitz maps and earthquakes.
The paper studies optimal maps between hyperbolic surfaces, focusing on their rigidity and obstructions.
problem Finding optimal Lipschitz maps between hyperbolic surfaces and understanding their rigidity and obstructions.
method Introducing deflations, optimal maps to trees that obstruct optimal maps between surfaces, and using a smooth orthogeodesic foliation.
result Deflations are the main obstructions to optimal maps between hyperbolic surfaces, and they are essentially the only ones.
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
We give asymptotic bounds for the optimal Lipschitz constants for the systole map from the Teichmuller space to the curve complex. We give similar results to those known for closed surfaces in the cases when the genus is fixed or the ratio of genus and punctures is a rational number.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüll…
Proves existence of curved surfaces in hyperbolic space.
problem Finding surfaces with specific curvature and boundary conditions.
method Proves existence using Weingarten curvature and asymptotic boundary conditions.
result Proves existence of locally Lipschitz continuous hypersurfaces.
The paper sharpens a theorem about surfaces with zero Gaussian curvature.
problem Quantifying the isometric property of surfaces with zero Gaussian curvature.
method Asymptotically sharp quantitative version of a classical theorem using isothermal coordinates.
result An isothermal coordinate map from a Riemannian disc to an Euclidean disc is bi-Lipschitz with a constant of exp(4ε).
We prove that Lipschitz intrinsic graphs in the Heisenberg groups Hn, with n>1, which are vanishing viscosity solutions of the minimal surface equation are smooth.
Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.
problem Lorentzian distances to Cauchy surfaces
method Conjectures based on Cauchy temporal functions
result Lorentz distances to Cauchy surfaces are not locally equi-Lipschitz
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
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 …
This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.
problem Analyzing the properties of maps between hyperbolic surfaces, particularly best Lipschitz maps and their relationship to geodesic laminations.
method The authors produce best Lipschitz maps as limits of minimizers of p-Schatten integrals, addressing existence and regularity issues.
result The support of the measure dv, the derivative of a Lie algebra valued function v, lies on the canonical geodesic lamination constructed by Thurston.
Two definitions for the rectfiability of hypersurfaces in Heisenberg groups Hn have been proposed: one based on H-regular surfaces, and the other on Lipschitz images of subsets of codimension-1 vertical subgroups. The equivalence between these notions remains an open problem. Recent partial res…
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…
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…
In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense o…
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'…
New limits of minimal surface systems have surprising large interior parts.
problem Minimal surface system limits with large interior vertical and non-minimal portions.
method Construction of limits with smallest possible dimension and codimension.
result Limits of minimal surface systems can have surprising large interior parts.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
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,…
The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.
problem Investigating maximal stretch and Lipschitz maps on negatively curved manifolds.
method Defined maximal stretch for negatively curved manifolds and connected it to best Lipschitz maps.
result The Mather set may not be lifts of geodesic laminations but shares similar features.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
Study 2D spaces with curvature, finding a graph structure.
problem Understanding the geometry of 2D spaces with curvature constraints.
method Analyzing spaces as unions of disks, identifying singular points.
result Obtained a graph structure of topological singular points.
We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spac…
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, w…
Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.
problem Existence time of the Willmore flow for various initial conditions.
method Established minimal existence time for Willmore flow using geometric data and conservation laws.
result Minimal existence time is a function of geometric data for general weak Lipschitz initial data.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for H-minimal surfaces in the Heisenberg group. method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
Paper proves DN map determination for simple surfaces with low regularity metrics.
problem Determining DN map from scattering relation for surfaces with low regularity metrics.
method Modified technical results and used microlocal analysis for metrics with finite regularity.
result Scattering relation determines DN map for C17 surfaces, and for C1,1 metrics using Lipschitz distance function. The paper shows how Hamiltonian diffeomorphisms and homeomorphisms can be broken down into smaller, manageable pieces.
problem Fragmenting Hamiltonian diffeomorphisms and homeomorphisms on surfaces.
method Develops a C0-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces, proving it with a Lipschitz estimate. result Hamiltonian diffeomorphisms and homeomorphisms can be decomposed into smaller, compactly supported pieces with a Lipschitz estimate on the C0-norm.