The study examines properties of universal covers of compact Kahler manifolds under Caratheodory measure hyperbolicity.
problem Understanding the properties of universal covers of compact Kahler manifolds under specific geometric conditions.
method Comparing invariant volume forms and using similar methods to establish inequalities.
result Established inequalities between the volume/restricted volume of canonical bundles and Caratheodory measure of universal covers/covering.
We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that C^1 submanifo…
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
New method improves scalability of SGD for large datasets.
problem High variance in stochastic gradient descent.
method Adaptive measure reduction with Carathéodory's theorem.
result Improved scalability to high-dimensional spaces.
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
Caratheodory's axiom limits arbitrage in resource-limited systems.
problem Non-arbitrage constraints in resource-limited financial systems.
method Preserving Caratheodory's axiom in resource-limited systems.
result Exponential family is the necessary geometric structure for both thermodynamics and finance.
Generalizes Carathéodory form for higher-order field theories.
problem Extending the Carathéodory form to second and higher-order Lagrangians.
method Geometric operations applied to the Poincaré--Cartan form and Lepage forms.
result Generalized Carathéodory form for second and higher-order Lagrangians.
Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.
problem Characterizing domains with specific metric properties.
method Analyzing Carathéodory and Bergman metrics on strictly pseudoconvex domains.
result Domains with specific metric properties are biholomorphically equivalent to balls.
We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-…
Maps preserving Carathéodory distance between symmetric domains are rigid.
problem Rigidity of maps preserving Carathéodory distance between bounded symmetric domains.
method Large-scale geometry of Carathéodory distance, horocompactification, Gromov product.
result Maps preserving Carathéodory distance are rigid and either holomorphic or antiholomorphic.
We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…
Different distances on symmetrical domains in complex space.
problem Comparing distances on specific complex domains.
method Examined Carathéodory pseudo-distance and Kähler-Einstein metric distances.
result Found the distances differ on certain complex domains.
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
Classifies geodesics for Carathéodory metric on Teichmüller spaces.
problem Distinguishing Carathéodory and Teichmüller metrics on Teichmüller disks.
method Dynamical results of Minsky, Smillie, and Weiss; complex-analytic criterion.
result Proves conjecture for specific surfaces, extending result to punctured surfaces.
Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.
problem Identifying geodesics in Heisenberg group under two metrics.
method Examined Heisenberg group H1 with Koranyi- and Carnot-Caratheodory metrics.
result Geodesics coincide for both metrics in Heisenberg group.
Extends Carathéodory's theorem to multidimensional domains with constant curvature.
problem Characterizing biholomorphic domains with constant holomorphic curvature.
method Using Bergman representative coordinates and Calabi's diastasis.
result Provides sufficient conditions for the boundary of a biholomorphic ball to be a topological sphere.
Study on Carathéodory metric discrepancies on Teichmüller spaces.
problem Determine if Carathéodory metric agrees with Teichmüller metric on Teichmüller spaces.
method Analyzed general case of Teichmüller spaces, proving metric discrepancies except for seven specific spaces.
result Proved Carathéodory metric disagrees with Teichmüller metric on all Teichmüller spaces except seven specific ones.
Extends strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces.
problem Proving strong comparison principle for p-harmonic functions in specific geometric settings.
method Extends Bony's propagation of support argument to C^1 solutions of sub-elliptic p-Laplacian.
result Proves strong maximum and comparison principles for p-harmonic functions.
Extends scaling maps theory to manifolds with boundary.
problem Quantitative study of Carnot-Carathéodory balls on manifolds with boundary.
method Introduction of scaling maps adapted to Carnot-Carathéodory balls and Hörmander vector fields on manifolds with boundary.
result First paper in a series studying maximally subelliptic boundary value problems.
Researchers describe horofunctions in noncompact Hermitian symmetric spaces.
problem Understanding horofunctions in noncompact Hermitian symmetric spaces.
method Realized noncompact Hermitian symmetric spaces as open unit balls in Banach spaces with Jordan structures.
result Complete description of horofunctions in the metric compactification.
New geometric proof shows index of umbilic points on analytic surfaces is at most one.
problem Proving the Carathéodory Conjecture for compact simply connected embedded surfaces.
method Geometric analysis of degenerate umbilic points on analytic surfaces.
result Index of an umbilic on an analytic surface cannot be an integer larger than one.
The paper studies metrics and geodesics on a quaternionic Heisenberg group.
problem Characterizing geodesics and distances on a quaternionic Heisenberg group.
method Defining and analyzing a sequence of Riemannian metrics, deriving formulas for mean curvature.
result Explicit description of Carnot-Carathéodory distance and spheres.
Researchers confirm a conjecture about metrics on a specific Teichmüller space.
problem Proving the conjecture about metrics on a specific Teichmüller space.
method Analyzing a specific Teichmüller space of genus 2 with 0 punctures.
result The conjecture is confirmed for a specific Teichmüller space.
Paper solves Carathéodory's conjecture for C2-regular convex surfaces.
problem Carathéodory's conjecture about convex surfaces.
method Index formula derived from Lorentz--Minkowski 4-space analysis.
result Affirmative solution to conjecture for C2-regular surfaces. The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.
problem Metric inequivalence and characterization of proper holomorphic maps.
method Explicit characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
result Characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we…
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
The paper examines convergence of distances in Lipschitz structures on manifolds.
problem Convergence of distances in Lipschitz vector fields and norms on manifolds.
method Analysis of convergence of distances associated to converging structures of Lipschitz vector fields and norms.
result Under mild controllability assumption, distances converge locally uniformly to the limit Carnot-Carathéodory distance.
For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanif…
We introduce a new class of unbounded model subdomains of C2 for the □b problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot-Carathéodory metric induced on the boundary of such domains by …
We find necessary and sufficient conditions for a Lipschitz map f:RE→X, into a metric space to have the image with the k-dimensional Hausdorff measure equal zero, Hk(f(E))=0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a …
New metrics contradicting old conjectures on umbilic points.
problem Contradicting old conjectures about umbilic points.
method Constructing Riemannian metrics arbitrarily close to flat metrics.
result Explicit construction of metrics with isolated umbilic points.
The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--C…
In this paper we study the invariant Carnot-Caratheodory metrics on SU(2)≃S3, SO(3) and SL(2) induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…
We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in C2. When the hypersurface has a uniform global structure, we show that a metric ball of radius δ≫1 either has volume on the order of δ3 or δ4. We also give necessary and …
This paper introduces Hausdorff measure and its applications in fractal geometry.
problem Defining and applying Hausdorff measure to fractal geometry.
method Definition of Hausdorff outer measure, Caratheodory's criterion, construction of Hausdorff measure, and introduction of Hausdorff dimension.
result Demonstrates the Hausdorff dimension of the Cantor ternary set.
The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
problem Analyzing holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
method Using Bochner formulas and comparison theorems.
result Established Schwarz type lemmas for holomorphic maps.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups. New findings on optimization landscape of Toeplitz covariance estimation.
problem Understanding the geometry of the Gaussian maximum-likelihood objective for Toeplitz covariance estimation.
method Overparameterized Carathéodory representation of positive definite Toeplitz covariance matrices, focusing on both amplitudes and frequencies.
result Joint optimization of amplitudes and frequencies leads to a benign population landscape, allowing for global recovery of the true Toeplitz covariance.
We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the norm…
Improved Sobolev mappings in Carnot groups with weaker assumptions.
problem Improving Sobolev mappings in Carnot groups with weaker conditions.
method Using Buser-Karcher center-of-mass and polynomial expressions in moments.
result Rigidity and structural results hold under weaker Sobolev exponents.
The abstract discusses financial irreversibility using quantum mechanics and projective geometry.
problem Financial irreversibility and its limitations in trading strategies.
method Projective geometry and Taylor expansion of directed distance in quantum systems.
result Fundamental asymmetry under state exchange is a key factor in financial irreversibility.
Study of perimeter measures in Heisenberg group with sub-Finsler metric.
problem Isoperimetric problem in sub-Finsler Heisenberg group.
method Reduction of Minkowski content to Lebesgue surface area, study of Finsler normed planes, use of CC-geodesics.
result Evidence supports Pansu's conjecture in sub-Finsler case, but with lower isoperimetric ratio.
An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetri…
A quadratic point on a surface in RP3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact non-degenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjectur…
Sobolev mappings preserve the Rumin complex on contact manifolds.
problem Preserving the Rumin complex under Sobolev mappings on contact manifolds.
method Using the Pullback Theorem, Pansu pullback is shown to induce chain mappings between Rumin complexes and de Rham complexes.
result The Rumin flat complex is bilipschitz invariant under Sobolev mappings between contact manifolds.