Complex functional maps link tangent bundles, preserving orientation and angles.
problem Linking tangent bundles for orientation-aware correspondence.
method Endow tangent bundles with complex structures to enable robust transfer of tangent vector fields.
result Establishes orientation-aware correspondence without relying on descriptors or extra regularization.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Estimates optimal transport maps with known cost functions.
problem Ensuring optimal transport maps correspond to real-world usefulness.
method Differentiable neural ground costs with known Monge map forms.
result General approach for incorporating prior information.
Extends Fatou theorem to bounded harmonic maps.
problem Classical Fatou theorem for bounded harmonic functions.
method Extending theorem to bounded harmonic maps.
result Identifies bounded harmonic maps on unit disk with bounded measurable functions on boundary.
The identity map of certain Einstein manifolds is stable in both energy and bienergy.
problem Stability of the identity map in Einstein manifolds.
method Investigation of conformal-biharmonic stability compared to harmonic stability.
result The conformal-biharmonic index coincides with the harmonic index, except for the 4D Euclidean sphere.
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 paper constructs gluing maps for harmonic maps between Riemannian manifolds.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
Classifies 2-uniform maps on torus with formulas and asymptotic bounds.
problem Classifying 2-uniform maps on torus.
method Classification through arithmetic functions and asymptotic analysis.
result Explicit formulas and asymptotic bounds for 2-uniform maps on torus.
Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In …
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…
We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappin…
Quaternionic analysis proves minimum of Willmore functional on Riemann surfaces.
problem Finding minimum of Willmore functional on Riemann surfaces.
method Extending quaternionic analysis to weakly conformal maps and using Darboux transformation.
result Minimum of Willmore functional on Riemann surfaces is achieved by weakly conformal maps.
Harmonic maps are described using Jacobi elliptic functions.
problem None explicitly stated; focuses on existing work.
method Use of Jacobi elliptic functions to describe harmonic maps.
result Harmonic maps can be described using Jacobi elliptic functions.
The author studies regions foliated by 1D families of functions and their applications.
problem Understanding regions represented as foliated forms and natural smooth maps onto them.
method Investigates natural smooth maps respecting canonical projections and moment maps, focusing on foliated regions.
result Discusses the 1st derivative of functions and critical sets in foliated regions.
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
Optimal transport for functional data using Hilbert-Schmidt operators.
problem Optimal transport for distributions on function spaces with partially represented stochastic maps.
method Regularization technique to restrict transport maps to Hilbert-Schmidt operators, developing an efficient algorithm.
result Existence, uniqueness, and consistency of the Hilbert-Schmidt operator estimate for the transport map.
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…
Locally maximizing orbits studied in twist maps and billiards.
problem Characterize orbits in locally maximizing class for twist maps.
method Geometric and variational analysis of orbits in the cotangent bundle of a torus or ball bundle over a sphere.
result Two generating functions for the Birkhoff billiard map have the same class of locally maximizing orbits.
Extends symphonic maps to bi-symphonic maps between Riemannian manifolds.
problem Defining and exploring new types of maps between Riemannian manifolds.
method Introduces bi-symphonic maps by analyzing the bi-energy functional.
result New types of maps (bi-symphonic) with associated bi-energy functional.
Paper constructs Thom-Smale complex using instantons from Morse functions.
problem Constructing Thom-Smale complex for Morse functions.
method Analytic instanton construction using eigenspaces of mapping cone Laplacian.
result Instanton complex is cochain isomorphic to Thom-Smale complex.
Stability of Dehn functions proven for ultralimits of Sobolev maps.
problem Stability of Dehn functions under ultralimit of Sobolev maps.
method Using ultralimits of Sobolev maps and properties of ultralimits of Lipschitz maps.
result Stability of Dehn functions under ultraconvergence of pointed length spaces.
This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, tho…
Defines manifolds of mappings between function spaces and discusses their properties.
problem Defining smooth manifolds of mappings between function spaces.
method Defines a smooth manifold structure on sets of continuous mappings and discusses properties of natural mappings.
result Properties of spaces of sections and smoothness of natural mappings between spaces of mappings.
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …
In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …
The paper explores α−harmonic maps and their stability, proving key properties and conditions.
problem Existence and stability of α−harmonic maps between Riemannian manifolds. method Analysis of α−energy functional, construction of α−harmonic maps, and stability conditions. result Conditions for the stability of α−harmonic maps and their instability from compact manifolds. Proves generalized Chen's conjecture for biharmonic maps on foliations.
problem Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
method Analyzes (F,F')-biharmonic maps and their critical points.
result Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
New length functions on mapping class groups linked to simplicial volumes of mapping tori.
problem Understanding the relationship between mapping class groups and simplicial volumes of mapping tori.
method Introducing filling volumes as length functions and proving their properties.
result Real filling volumes equal the simplicial volume of mapping tori, while integral filling volumes are not smaller than the stable integral simplicial volume.
It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…
A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a me…
Derives functional Itô formula for non-anticipative maps of rough paths.
problem Functional Itô formula for non-anticipative maps of càdlàg rough paths.
method Approximation properties of the signature and Marcus transformation.
result Functional Taylor expansion for sufficiently regular non-anticipative maps.
New algorithm speeds up online mapping of unknown terrains.
problem Increasing computational demands of GP mapping as area expands.
method Recursive GP mapping using local basis functions in an information filter.
result Reduces overall computational complexity and speeds up mapping.
The study restricts manifolds with certain explicit SGL maps and constructs them.
problem Restrictions on manifolds admitting specific SGL maps.
method Generalization of Morse functions and canonical projections to construct SGL maps.
result Manifolds admitting certain explicit SGL maps are strongly topologically restricted.
Universal approximation theorem for differentiable maps on infinite-dimensional manifolds
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
Local constancy of index for certain gradient mappings proved.
problem Proving the local constancy of the index for specific gradient mappings.
method Using a more general theorem for quasiregular gradient mappings, deducing the result from the Hessian's properties.
result The index is locally constant for C1,1 functions with uniformly positive determinant Hessian almost everywhere. A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…
The paper studies f-polyharmonic maps and their properties.
problem Understanding and characterizing f-polyharmonic maps. method Deriving the Euler-Lagrange equation and analyzing specific cases.
result Every f-polyharmonic function on a closed Riemannian manifold is constant. The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. Researchers compute dimensions of GLN-skein modules for genus-one mapping tori.
problem Computing dimensions of GLN-skein modules for mapping tori.
method Explicit Euler product expansion of the skein partition function.
result Explicit computation of dimensions and generating function.
We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in both Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.
Study on generalized ξ-parallel maps in Riemannian geometry.
problem Characterizing and understanding generalized ξ-parallel maps.
method Defined energy functional, derived first variation formula, and Euler-Lagrange equation.
result Established fundamental properties and relationships with harmonic and biharmonic maps.
Study moment maps coupled with convex functions to find critical points.
problem Understanding critical points of moment maps coupled with convex functions.
method Develop a theory of moment maps coupled with an Ad_K-invariant convex function f on k*.
result Interpret Kähler-Ricci solitons as a special case of generalized extremal metrics.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Study connects Gaussian processes and regularization for sequence-function mappings.
problem Understanding and interpreting sequence-function maps in biology.
method Relates Gaussian process priors, regularization, and gauge fixing in overparameterized weight space.
result Established the relationship between regularized regression and Gaussian processes in function space.
Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.
problem Quantifying uniqueness of conformal-harmonic maps on 4-manifolds.
method Proves a quantitative uniqueness result using convexity and second order Hardy inequality.
result Proves a version of second order Hardy inequality on manifolds.