Constructs harmonic maps between special geometric shapes.
problem Creating harmonic maps between specific types of geometric shapes.
method Equivariant harmonic maps constructed between cohomogeneity one manifolds.
result Developed a method to construct harmonic maps.
The paper defines metallic maps and explores their properties and constancy.
problem Exploring properties and constancy of metallic maps between Riemannian manifolds.
method Introducing metallic maps, providing conditions for total geodesy and harmonicity, investigating constancy under holomorphic-like conditions.
result Certain metallic maps are constant under specific conditions.
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…
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.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …
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.
Study generalizes map properties between Hermitian manifolds preserving specific forms.
problem Understanding maps between Hermitian manifolds that preserve certain forms.
method Generalizing results from Chan-Yuan [2025] to new maps.
result Obtained further rigidity and non-existence theorems.
Paper derives formulas for holomorphic maps between Hermitian manifolds and proves related theorems.
problem Holomorphic maps between Hermitian manifolds and their properties.
method Derives ∂∂-Bochner formulas and proves Schwarz lemma type estimates. result Generalizes Ni's results to Hermitian manifolds, proving rigidity and degeneracy theorems.
Compactness theorem for quasiregular maps between manifolds.
problem Compactness of sequences of quasiregular mappings.
method Gromov's compactness theorem for pseudoholomorphic curves.
result Subsequence of quasiregular mappings converges to a quasiregular map on a nodal manifold.
Study on harmonicity of maps between different types of almost contact metric manifolds.
problem Understanding harmonicity of maps between various almost contact metric manifolds.
method Analyzing and deriving new results for different subclasses of almost contact metric manifolds.
result Obtained new results and recovered, generalized, and corrected known results.
New functional interpolates harmonic and biharmonic maps.
problem Interpolating between harmonic and biharmonic maps.
method Introduced a new action functional for Riemannian maps.
result Initiated rigorous mathematical treatment of the new functional.
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.
First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th century, but maps between surfaces kept to be an active topic in the 20th centur…
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…
New statistical biharmonic maps derived from a variation problem.
problem Variation problem for mappings between statistical manifolds.
method Statistical biharmonic maps derived from the Euler-Lagrange equation.
result Improper affine hyperspheres induce examples of statistical biharmonic maps.
We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killin…
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain …
The paper establishes Schwarz type lemmas for pseudo-Hermitian manifolds.
problem Understanding the geometry and mappings of pseudo-Hermitian manifolds.
method Using sub-Laplacian or Hessian type Bochner formulas and comparison theorems.
result Established Schwarz type results for \emph{CR} maps and transversally holomorphic maps.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
In this short survey we report on the theory of biharmonic maps between Riemannian manifolds.
We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian …
Notes on quasiregular maps between Riemannian manifolds, preserving Sobolev forms.
problem Extending quasiregular map theory from Euclidean to Riemannian manifolds.
method Recalling different approaches to first-order Sobolev spaces, showing equivalence, and transferring key theorems.
result Pull-backs with quasiregular maps preserve Sobolev differential forms of the conformal exponent.
The purpose of this paper is to study the harmonicity of maps to or from para-Sasakian manifolds. We derive the condition for the tension field of paraholomorphic map between almost para-Hermitian manifold and para-Sasakian manifold. The necessary and sufficient condition for a paraholomorphic map between para-Sasakian…
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.
Paper studies geodesic and harmonic mappings, solving inverse problems.
problem Relation between geodesic and harmonic mappings.
method Calculus of variations on fibred manifolds to solve inverse problems.
result Found that the connection on the source manifold need not be metric.
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …
Smooth maps between manifolds form a Banach manifold.
problem Understanding the structure of maps between manifolds.
method Detailed rigorous proof of the smooth manifold structure.
result The set of k times continuously differentiable maps between manifolds forms a smooth Banach manifold. Study pseudoholomorphic maps using canonical connection.
problem Characterize pseudoholomorphic maps between almost Hermitian manifolds.
method Use canonical connection and Bochner formulas to derive estimates and theorems.
result Obtained C2-estimate of canonical second fundamental form and Liouville type theorems. The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
Maps between positively curved manifolds with non-increasing area are rigid.
problem Understanding maps between manifolds with positive curvature and non-increasing area.
method Exploring the graphical mean curvature flow and using Brendle's sphere theorem.
result Maps between certain positively curved manifolds are homotopy trivial, Riemannian submersion, local isometry, or isometric immersion.
We study the J-flow on the toric manifolds, through study the transition map between the moment maps induced by two Kähler metrics, which is a diffeomorphism between polytopes. This is similar to the work of Fang-Lai, under the assumption of Calabi symmetry, they study the monotone map between two intervals. We get a p…
The paper examines HM-tensional and HS-tensional maps between Riemannian manifolds.
problem Analyzing tension fields of maps between Riemannian manifolds.
method Investigating harmonic maps and harmonic sections as tension fields.
result Characterization and properties of HM-tensional and HS-tensional maps. Analyzes smoothness and classification of maps between manifolds.
problem Analyzing interpolating sesqui-harmonic maps between Riemannian manifolds.
method Derives a conservation law and uses it to show smoothness of weak solutions; obtains classification results.
result Smoothness of weak solutions and classification results for interpolating sesqui-harmonic maps.
Polyharmonic maps are harmonic under specific conditions.
problem Conditions for polyharmonic maps to be harmonic.
method Proving polyharmonic maps are harmonic under smallness and integrability conditions.
result Polyharmonic maps are harmonic under certain conditions.
We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. `conformal hypercomplex' manifolds) and quaternionic manifolds of 1 dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between th…
Notes on harmonic maps between manifolds, existence and regularity covered.
problem Existence and regularity of harmonic maps between Riemannian manifolds.
method Lecture-based approach covering harmonic maps, pluriharmonic maps, and related theorems.
result Coverage of existence and regularity of harmonic maps, including Siu-Sampson formula and Donaldson-Corlette theorem.
Harmonic maps prove quasi-isometric embeddings are close to unique.
problem Understanding quasi-isometric embeddings between pinched Hadamard manifolds.
method Proving quasi-isometric maps are close to harmonic maps.
result Quasi-isometric maps are within bounded distance from a unique harmonic map.
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
The study proves unique continuation for biharmonic maps.
problem Unique continuation of biharmonic maps between manifolds.
method Proof of unique continuation results.
result Proves several unique continuation results for biharmonic maps.
The paper studies convergence of discrete harmonic maps to smooth ones.
problem Discretization of harmonic maps between Riemannian manifolds.
method Introducing triangulations with vertex and edge weights, and studying convergence conditions.
result Suitable conditions on weighted triangulations ensure convergence of discrete harmonic maps to smooth ones.
In this paper we introduce a natural definition for the affine maps between two Finsler manifolds (M,F) and (N,F~) and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;…
Minimal simplicial maps constructed for spheres and manifolds.
problem Constructing minimal simplicial maps of specific degrees.
method Triangulations and degree constructions for manifolds and spheres.
result Minimal triangulations for degree d self-maps of Sn−1imesS1. Geometric inequality linking Dirichlet and bienergy for maps between Riemannian manifolds.
problem Relating Dirichlet and bienergy for maps between Riemannian manifolds.
method Established a geometric inequality relating the Dirichlet energy and bienergy of smooth maps between Riemannian manifolds.
result Proved that E2(f)≥RicminE1(f) under specified conditions. A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. …
The paper proves rigidity theorems for maps between Riemannian manifolds.
problem Rigidity of maps between Riemannian manifolds.
method Generalizes Piola identity and uses Lp convergence. result Proves the existence of isometric immersions under certain conditions.