Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. Study cohomology classes related to harmonic maps on submersions.
problem Understanding harmonic maps on submersions and their cohomology.
method Extending previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic and f-harmonic maps.
result Extend results on Riemannian submersions to F-harmonic and f-harmonic maps.
The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.
problem Existence of harmonic and bi-harmonic maps into certain Riemannian manifolds.
method Analysis of manifolds with conformal vector fields or Ricci solitons.
result Nonexistence of harmonic and bi-harmonic maps in specified conditions.
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.
Extends p-harmonic map theory for new properties.
problem No specific problem stated; extends existing theory.
method Extended p-harmonic and biharmonic map definitions.
result New properties of generalized stable p-harmonic maps.
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
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.
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 …
∞-Harmonic maps are a generalization of ∞-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞-harmonic maps from and into a sphere, quadratic ∞-harmonic maps between E…
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 proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Study cohomology classes related to n-harmonic morphisms and F-harmonic maps.
problem Understanding cohomology classes associated with n-harmonic morphisms and F-harmonic maps. method Utilizing the n-conservation law (2.6) to obtain sharp results. result Sharp results on cohomology classes related to n-harmonic morphisms and F-harmonic maps. Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
Stability for ΦS,F,H harmonic map and ΦT,F,H harmonic mapmath.DG The paper examines stability of harmonic maps on specific manifolds.
problem Stability of harmonic maps on compact convex hypersurfaces.
method Analyzes stability conditions for ΦS,F,H and ΦT,F,H harmonic maps. result Provides theorems to determine stability of ΦS,F,H and ΦT,F,H harmonic maps. The paper studies geometric properties of Φ(3)-harmonic maps and proves Liouville type results.
problem Exploring geometric properties of Φ(3)-harmonic maps. method Unified geometric analytic methods, first and second variation formulas, stress-energy tensor, conservation law, monotonicity formula, asymptotic assumption, extrinsic average variational method.
result Proves Liouville type results for Φ(3)-harmonic maps. J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existe…
α-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to α-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For α>1, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in …
Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.
problem Characterizing the qualitative behavior of 4-harmonic and ES-4-harmonic maps.
method Proving triviality of finite energy solutions for both maps.
result Finite energy solutions of both 4-harmonic and ES-4-harmonic maps are trivial.
The paper examines stability of subelliptic harmonic maps with potential.
problem Stability of subelliptic harmonic maps with potential.
method Derived first and second variation formulas, proved stability conditions, and gave instability results.
result Subelliptic harmonic maps with potential are stable under certain curvature and potential conditions.
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 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. We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of p% -harmonic maps as p→∞. Infinity harmoncity appears in many familiar contexts. For example,…
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.
The paper proves a Liouville theorem for a specific type of harmonic maps on foliated manifolds.
problem Investigating harmonic maps on foliated Riemannian manifolds.
method First variational formulas, generalized Weitzenböck type formula, and Liouville type theorem for (F,F′)p-harmonic maps. result Established a Liouville type theorem for (F,F′)p-harmonic maps. Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Study on stability of harmonic maps with sub-Riemannian geometry.
problem Stability of exponentially subelliptic harmonic maps.
method Derived first and second variation formulas, applied to prove stability under certain conditions.
result Exponentially subelliptic harmonic maps are stable if the target manifold has nonpositive curvature.
The paper studies harmonic identity maps on Riemannian manifolds.
problem Understanding harmonicity of identity maps on Riemannian manifolds.
method Constructing new examples and defining a symmetric tensor field.
result New examples of identity harmonic maps are constructed.
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
Sharp inequality for p-harmonic maps with new optimal constant.
problem Deriving the sharp vectorial Kato inequality for p-harmonic mappings. method Analyzing the inequality for p-harmonic mappings and comparing with scalar valued cases. result Established the optimal constant for p-harmonic maps and enhanced the range of p values for regularity. Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
The paper characterizes harmonic maps and studies their properties on Riemannian manifolds.
problem Characterizing and analyzing harmonic maps between Riemannian manifolds.
method Analytic and geometric methods, including L2-orthogonal decomposition and energy density analysis.
result A criterion for harmonic submersions and diffeomorphisms, and new results linking harmonic symmetric bilinear forms and metrics.
New ε-harmonic maps of low degree are rigid under certain energy bounds.
problem Understanding the rigidity of ε-harmonic maps of low degree. method Analysis of ε-harmonic maps and their critical points. result Non-trivial ε-harmonic maps of degree zero exist with energy above 8π. Study proves existence of non-trivial harmonic map flows to hemispheres.
problem Existence of non-trivial harmonic map flows to hemispheres.
method Construction of infinitely many weak solutions to harmonic map flow starting from non-minimizing but stationary maps.
result Proves existence of non-trivial self-expanding harmonic map flows to hemispheres.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
New rigidity estimate derived via harmonic map flow.
problem Rigidity of maps from S2 to S2. method Harmonic map flow approach.
result Rigidity estimate derived.
In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.
J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k"…
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.
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. Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
Extends harmonic map theory to arbitrary surfaces.
problem Classifying harmonic maps from arbitrary surfaces.
method New theory of nilpotent cycles from diagrams, criterion for uniton number finiteness.
result New classification of minimal surfaces and constancy of finite type harmonic maps.
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.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps. Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
Unified study of harmonic maps between pseudo-Riemannian surfaces.
problem Classifying harmonic maps between pseudo-Riemannian surfaces.
method Unified formalism and Bäcklund transformation.
result Unified solutions to harmonic map equations and corresponding maps.