Study intrinsic regular surfaces in Carnot groups, generalizing results from Heisenberg groups.
problem Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
method Generalize results from Heisenberg groups to Carnot groups.
result Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
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.
This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.
problem Boundedness of geodesic curvature measures near cross cap singularities.
method Analyzes intrinsic cross cap singularities and extends Gauss-Bonnet formula.
result Proves boundedness of geodesic curvature measures for curves near cross cap singularities.
This paper proves metric rectifiability for certain Heisenberg group surfaces.
problem Equivalence of rectifiability definitions for Heisenberg group surfaces.
method New criterion for finding bilipschitz maps between metric spaces.
result Metric bilipschitz rectifiability for α-Hölder continuous surfaces in Hn. We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform ε−regularity estimates whic…
Convex surfaces derived from specific Riemannian manifolds with high regularity.
problem Proving convexity of surfaces derived from Riemannian manifolds.
method Analyzing solutions to the very weak Monge-Ampère equation.
result Proved convexity of weakly regular surfaces with nonnegative intrinsic curvature.
Two definitions quantify C2,α regularity of Riemannian surfaces.
problem Quantify the regularity of Riemannian surfaces.
method Intrinsic and extrinsic definitions using Hölder norms and smooth local representations.
result Intrinsic and extrinsic definitions are equivalent up to a constant.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type: one in terms of Lie brackets of a complex tangent v…
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
Study introduces weak elastic energy for curves on Riemannian surfaces.
problem Detecting curvature of curves on Riemannian surfaces.
method Relaxation starting from inscribed geodesic polygonals, defined in normalized isothermal coordinates.
result Relaxed energy detects intrinsic second-order Sobolev regularity and agrees with geodesic curvature.
We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global express…
Defines new metric space sections with Ahlfors-David regularity.
problem Defining and analyzing new types of sections in metric spaces.
method Introducing intrinsically quasi-symmetric sections and proving their Ahlfors-David regularity.
result Proves Ahlfors-David regularity for intrinsically quasi-symmetric sections.
A new method evolves point clouds using B-splines for smooth surfaces.
problem Evolution of smooth surfaces from discrete point clouds.
method Adaptive Lagrangian B-spline framework for geometric evolution.
result Efficient and accurate reproduction of surface evolution phenomena.
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and t…
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call…
We propose a new method for estimating the intrinsic dimension of a dataset by applying the principle of regularized maximum likelihood to the distances between close neighbors. We propose a regularization scheme which is motivated by divergence minimization principles. We derive the estimator by a Poisson process appr…
We consider surfaces of class C1 in the 3-dimensional sub-Riemannian Heisenberg group H1. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…
The paper characterizes biconservative surfaces in hyperbolic 4-space.
problem Characterizing biconservative surfaces in hyperbolic 4-space.
method Establishing intrinsic conditions and analyzing geometric properties.
result Biconservative surfaces in hyperbolic 4-space satisfy a specific intrinsic condition.
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.
New surfaces in Lorentz-Minkowski space with constant mean curvature identified.
problem Identifying surfaces with constant mean curvature in Lorentz-Minkowski space.
method Using a specific coordinate system and properties of the Weingarten endomorphism, the mean curvature is shown to be constant under certain conditions.
result Constant mean curvature surfaces identified, including spacelike and timelike Enneper surfaces.
Study on curves in Riemannian surfaces, focusing on total intrinsic curvature.
problem Understanding the total intrinsic curvature of irregular curves in Riemannian surfaces.
method Weak notion of parallel transport, bounded variation of angle, energy functional analysis.
result Total intrinsic curvature of irregular curves matches an energy functional.
Translation surfaces in Heisenberg group classified by Gauss map determinant.
problem Classifying translation surfaces with zero intrinsic curvature in Heisenberg group.
method Constructed surfaces as product of planar curves, classified by Gauss map determinant.
result Surfaces with vanishing intrinsic curvature identified.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
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.
Catenaries defined on any Riemannian surface using intrinsic distance.
problem Defining catenaries on Riemannian surfaces.
method Defining catenaries as critical points of a potential functional, calculating potential with intrinsic distance, and characterizing using curvature.
result Characterization of catenaries on various Riemannian surfaces.
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
Introduces intrinsically Lipschitz graphs in metric spaces.
problem Graphs in metric spaces with Lipschitz conditions.
method Focuses on quotient maps and intrinsically Lipschitz sections.
result Compactness, Ahlfors regularity, and extension theorems.
Defines intrinsically Hölder sections in metric spaces.
problem Characterizing Hölder sections in metric spaces.
method Introducing intrinsically Hölder graphs, proving compactness, regularity, and extension theorems.
result Establishes properties for intrinsically Hölder graphs, including vector space, convex set, and equivalence relation.
Solves Plateau's Problem in Heisenberg group for graphs.
problem Plateau's Problem in the Heisenberg group for intrinsic graphs.
method Geometric construction and calibration argument.
result Solves Plateau's Problem under smallness conditions.
Lectures on surface evolution through singularities.
problem Analyzing the mean curvature flow of surfaces and their singularities.
method Analysis of neck and conical singularities, using monotonicity formulas, epsilon-regularity, weak solutions, and blowup techniques.
result Unique evolution through neck singularities, nonuniqueness through conical singularities.
Maximal metric spheres found, related to Sobolev-to-Lipschitz property.
problem Finding maximal metric spheres.
method Characterizing maximal spheres by Sobolev-to-Lipschitz property.
result Maximal spheres uniquely characterized by Sobolev-to-Lipschitz property.
A new method integrates autoencoders with geometry regularization for manifold learning.
problem Extracting simplified low-dimensional representations that capture intrinsic geometry in data.
method Integrates autoencoders with a geometric regularization term based on diffusion potential distances.
result The method preserves intrinsic structure, enables out-of-sample extension, and faithful reconstruction.
We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp Ln−1-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…
The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…
Minimal surfaces' area bounds proven equivalent, extending known results.
problem Equivalence of area bounds for minimal surfaces.
method Combining recent breakthroughs, extending known results.
result Equivalence of intrinsic and extrinsic area density bounds for minimal immersions.
The study classifies constant mean curvature surfaces in curved spaces.
problem Classifying constant mean curvature surfaces in curved spaces.
method Analyzes constant mean curvature isometric immersions into S2imesR and H2imesR. result Provides new classifications of constant mean curvature surfaces in various curved spaces.
We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize len…
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.
Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.
problem Characterizing quasi-isometric embeddings of maps from cusped surfaces into moduli space.
method Investigates shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces, considering quasi-isometric embeddings with respect to Teichmüller distance and intrinsic distance.
result Characterizations of quasi-isometric embeddings are solely determined by the map's monodromy under mild conditions.
In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of disc…
The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.
problem Understanding properties of sections in metric spaces.
method Definition and investigation of intrinsically quasi-isometric sections in metric spaces.
result Properties of sections, including Ahlfors-David regularity and convexity, are defined and investigated.
We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.
We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov.