The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
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.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
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.
New type of harmonic Finsler manifolds created.
problem Creating new types of harmonic Finsler manifolds.
method Defined by a Riemannian metric and a special 1-form.
result Constructed harmonic and asymptotically harmonic Finsler manifolds of (α,β)-type. 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.
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. The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci provided examples showing that in the noncompact case the conjecture is wrong. H…
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. Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.
problem Characterizing compact RCD spaces as harmonic manifolds.
method Analyzing heat kernel and geodesic ball volumes for harmonicity.
result Compact RCD spaces are isometric to smooth manifolds under given conditions.
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.
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.
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 investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). 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 study examines harmonic forms on almost Hermitian manifolds and complex surfaces.
problem Analyzing harmonic forms on almost Hermitian manifolds and complex surfaces.
method Using techniques from Bott-Chern and Aeppli numbers, the study generalizes harmonic forms from complex and symplectic manifolds to almost Hermitian manifolds.
result Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold.
Corrected proof for 3D harmonic manifolds with minimal horospheres.
problem Proving 3D harmonic manifolds with minimal horospheres are either flat or hyperbolic.
method Provided a corrected proof for the classification of 3D harmonic manifolds.
result Classification of 3D harmonic manifolds: flat or hyperbolic.
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.
Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.
problem Classifying polynomial growth solutions to drift-harmonic equations on specific types of manifolds.
method Inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
result All drift-harmonic functions with polynomial growth asymptotically separate variables and dimensions of spaces are computed.
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …
In general, the product of harmonic forms is not harmonic. We study the top exterior power of harmonic two-forms on compact Kaehler manifolds. Often, it is not harmonic. This phenomenon is related to the geometry of the manifold and to the existence of rational curves in particular. K3 surfaces and hyperkaehler manifol…
The paper examines metrics on foliated manifolds that have special geometric properties.
problem Characterizing metrics on foliated manifolds with specific harmonic properties.
method Examining the properties of bundle-like metrics on foliated manifolds.
result The interior product of basic harmonic forms is basic harmonic under certain conditions.
Study on harmonicity of complex structure on product of trans-Sasakian manifolds.
problem Investigating harmonicity of complex structure on product of trans-Sasakian manifolds.
method Analysis of Levi-Civita connection and conditions for harmonicity on product manifold.
result Conditions for harmonicity of complex structure on product manifold of trans-Sasakian manifolds.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Paper proves energy identity and no-neck property for special harmonic maps.
problem Analyzing special harmonic maps with homogeneous targets.
method Introduced equivariant embedding for ε-harmonic case. result Energy identity and no-neck property established for ε- and α-harmonic maps. The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
Proves harmonicity equivalence on manifold metrics.
problem Harmonicity of metrics on pseudo-Riemannian manifolds.
method Equivalence of harmonicity on manifold metrics and their lifts.
result Harmonicity of a metric is equivalent to its lifts' harmonicity.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
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. Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of…
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.
problem Bounding the volume entropy of harmonic manifolds of hypergeometric type.
method Normalized Ricci curvature and entropy analysis.
result Upper and lower bounds for volume entropy of harmonic manifolds of hypergeometric type.
In this note we shall show that the sectional curvature of a harmonic manifold is bounded on both sides. In fact we shall give a pinching constant for all harmonic manifolds. We shall use the imbedding theorem for harmonic manifolds proved by Z.I.Szabo and the description of screw lines in hilbert spaces to prove the r…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of …
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. Paper studies heat flow for VT harmonic maps on compact manifolds.
problem Existence of VT harmonic maps and geodesics on compact manifolds.
method Heat flow method to solve Dirichlet problem and existence of geodesics.
result Existence of VT harmonic maps and geodesics under certain conditions.
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…
Study harmonicity of normal almost contact structures on Riemannian manifolds.
problem Understanding harmonicity of normal almost contact structures.
method Analyzing harmonicity through associated sections of a twistor bundle and rewriting equations in terms of curvature tensor.
result Conditions relating harmonicity of almost contact metric and almost complex structures.
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold…
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.