Sharp inequalities and extremizers for J functional on Kähler manifolds.
problem Understanding the large scale asymptotic of the J functional on Kähler metrics.
method Proving sharpness of inequalities and studying extremizing potentials/rays on toric Kähler manifolds, and existence of radial extremizers on general Kähler manifolds.
result Sharpness of inequalities and existence of extremizing potentials/rays on toric Kähler manifolds, and equivalence with plurisupported currents on general Kähler manifolds.
Study shows convergence of Fubini-Study currents to equilibrium metrics on Kähler manifolds.
problem Convergence of Fubini-Study currents to equilibrium metrics in Kähler geometry.
method Analysis of continuous Hermitian metrics and their Fubini-Study currents on line bundles.
result The scaled difference between Fubini-Study currents and equilibrium metrics converges to zero in the sense of currents.
We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors. We present current group extensions integrating some of these current Lie algebra extensions.
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
Proves unique continuation for area minimizing currents.
problem Ensuring area minimizing currents match minimal surfaces.
method Analyzes infinite order contact between currents and minimal surfaces.
result Currents and minimal surfaces coincide in a neighborhood.
Constructs brane current algebras from QP-manifolds, generalizing string currents.
problem Constructing brane current algebras from QP-manifolds.
method Using Poisson algebra and QP-manifolds (symplectic L∞-algebroids), the paper derives a universal geometric form for Poisson brackets of brane currents. result Derives a universal expression for 't Hooft anomaly in the presence of fluxes.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
Motivated by advantages of current-mode design, this brief contribution explores the implementation of weight matrices in neuromemristive systems via current-mode memristor crossbar circuits. After deriving theoretical results for the range and distribution of weights in the current-mode design, it is shown that any we…
Introduces Lie group actions in smoothing processes for currents and spaces with curvature.
problem Regularization of currents and metrics on manifolds and spaces with curvature.
method Actions of compact Lie groups in De Rham approximation and smoothing of Riemannian metrics.
result Effective smoothing processes for currents and metrics on manifolds and spaces with curvature.
To a tropical p-cycle VT in Rn, we naturally associate a normal closed and (p,p)-dimensional current on (C∗)n denoted by Tnp(VT). Such a "tropical current" Tnp(VT) will not be an integration current along any analytic set, si…
Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.
problem Proving denseness of rational currents on cusped hyperbolic surfaces.
method Using geodesic currents and subset currents, proving denseness through examples and continuous extension.
result Denseness of rational currents on cusped hyperbolic surfaces, including geodesics connecting cusps.
New characterization of geodesic currents via curve functionals.
problem Characterize geodesic currents using curve functionals.
method Purely axiomatic and combinatorial approach.
result Characterization of curve functionals dual to geodesic currents.
The paper connects bundle curvature to random zero currents.
problem Understanding the relationship between bundle curvature and random zero currents.
method Heat flow on Hermitian line bundles over Riemannian manifolds.
result Random zero currents connect bundle curvature to ground state zero current.
We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any N≥2 we provide examples of N-dimensional normal currents whose associated vector fields are simple, and whose supports are purely 2-unrectifiable and have Nagata dimension N. We show that in l∞ norm…
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theo…
New insights into currents of Hitchin representations with combinatorial restrictions.
problem Understanding currents associated with Hitchin representations.
method Defining dual spaces and analyzing combinatorial restrictions on self-intersection.
result Dual spaces of discrete boundary currents are polyhedral complexes with dimension at most n-1.
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
New definition of metric current yields Finsler geometry volume densities.
problem Defining volume functionals from Finsler geometry.
method Proposed a new definition of metric current and showed its utility.
result Obtained a family of extendibly convex volume densities.
We consider the problem of identifying current coupons for Agency backed To-be-Announced (TBA) Mortgage Backed Securities. In a doubly stochastic factor based model which allows for prepayment intensities to depend upon current and origination mortgage rates, as well as underlying investment factors, we identify the cu…
Constructs currents and heights on K3 surfaces.
problem Understanding the geometry and arithmetic of K3 surfaces.
method Constructs canonical positive currents and heights on K3 surfaces, equivariant for automorphism group.
result Continuous family of currents and heights defined over an enlarged boundary of the ample cone.
We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current T, we show that if the module X(∥T∥) of Weaver derivations is finitely generated, then T can be represented in terms of derivations; this extends previous results of Wi…
We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…
Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise t…
New currents derived from Killing-Yano tensors for gravity.
problem Finding new conserved currents in gravity.
method Using relations involving Riemann, Ricci, and Einstein tensors to introduce novel conserved currents.
result New currents derived from Killing-Yano tensors and their implications for conserved charges.
We take the novel perspective to view data not as a probability distribution but rather as a current. Primarily studied in the field of geometric measure theory, k-currents are continuous linear functionals acting on compactly supported smooth differential forms and can be understood as a generalized notion of orient…
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
We show that symmetries and gauge symmetries of a large class of 2-dimensional sigma models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the sigma model. We compute the current-current commutator and analyse the anomaly cancell…
Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.
problem Understanding the dual spaces of geodesic currents on hyperbolic surfaces.
method Analyzing the geometric properties of dual spaces, including their hyperbolicity and completeness.
result The dual spaces of geodesic currents are Gromov hyperbolic metric tree-graded spaces.
This paper aims to define and study currents and slices of currents in the Heisenberg group Hn. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define t…
Study b-divisors on Kähler manifolds linking them to currents.
problem Intersection theory of b-divisors on Kähler manifolds.
method Established correspondence between closed positive currents and nef b-divisors.
result Intersection theory of nef b-divisors answered.
Currents in higher dimensions can be reconstructed from projections.
problem Reconstructing currents from their projections in higher dimensions.
method Inversion formula for the exterior k-plane transform. result Currents in Rn can be reconstructed from their projections onto Rk. We study the properties of geodesic currents on free groups, particularly the "intersection form" that is similar to Bonahon's notion of the intersection number between geodesic currents on hyperbolic surfaces.
The paper proves existence and partial regularity for Legendrian area-minimizing currents.
problem Existence and partial regularity of Legendrian area-minimizing currents.
method Local minimization and application to the Legendrian Plateau problem.
result Existence and partial regularity of solutions to the Legendrian Plateau problem.
Integral currents with boundary of finite mass are integral.
problem Integral currents with boundary of finite mass are integral.
method De Giorgi's structure theorem for integer-valued BV functions and a cylindrical projection argument. result Integral currents with boundary of finite mass are integral.
Study improves boundary smoothness for area-minimizing currents with complex boundaries.
problem Boundary regularity for area-minimizing currents with arbitrary multiplicity.
method Generalization of Allard's boundary regularity theorem.
result Derivation of structural consequences from the generalized theorem.
Constructs solutions to Einstein-Maxwell-current system using Sasakian manifolds.
problem Solving the Einstein-Maxwell-Current system with inhomogeneous charged particle density.
method Using Sasakian manifolds to specify magnetic field and electric current.
result Solutions with arbitrary function describing charged particle density and curvature.
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to …
In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current w…
Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the…
In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a n-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the k-Hessian measures are ca…
Study on flat singularities of area-minimizing currents in codimension one.
problem Understanding flat singularities of area-minimizing currents in codimension one.
method Analyzing the structure of two-dimensional mod(q) area-minimizing currents near flat singularities.
result Currents are C1,α-perturbations of radially homogeneous special multiple-valued functions. Let L be a holomorphic line bundle over a compact Kähler manifold X endowed with a singular Hermitian metric h with curvature current c1(L,h)≥0. In certain cases when the wedge product c1(L,h)k is a well defined current for some positive integer k≤dimX, we prove that c1(L,h)k can be approxima…
Geodesic currents in strongly hyperbolic spaces are dense.
problem Characterizing geodesic currents with strongly hyperbolic dual pseudometrics.
method Combining finite-cover argument and boundary data characterization.
result Dense subset of geodesic currents with strongly hyperbolic dual pseudometrics.
A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.