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. 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.
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.
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. In this paper, we study compact generalized τ-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized τ-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for it. In the second part, we obtain some rigidity results for compact (τ,ρ)-qu…
We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called {\em pseudoharmonic morphisms}, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Final…
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. 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 study explores harmonic vector fields on a specific type of Riemannian Lie group.
problem Characterizing harmonic vector fields on a warped product of a line and a 3D Riemannian Lie group.
method Using a characteristic variational condition, the study applies to the case of a 3D Riemannian Lie group equipped with a left-invariant metric.
result Examples of harmonic vector fields on the warped product that are not left-invariant are provided.
The paper characterizes vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
problem Characterizing vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
method Characterization theorem and critical point condition for interpolating sesqui-harmonic vector fields.
result Conditions for vector fields to be interpolating sesqui-harmonic maps on compact manifolds.
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.
Study characterizes G2-structures on specific Lie groups and identifies harmonic conditions.
problem Characterizing and identifying harmonic G2-structures on almost Abelian Lie groups. method Analyzing left-invariant G2-structures, characterizing torsion forms, and using algebraic conditions. result Established algebraic conditions for harmonic G2-structures and identified admissible torsion classes. Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.
problem Characterizing solitons and their dual forms.
method Necessary and sufficient conditions for the dual form to be harmonic or Ricci harmonic are derived.
result Conditions for the dual form of a soliton to be harmonic or Ricci harmonic are provided.
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. 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.
We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition
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.
We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
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.
We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
New formulas for Bertrand curves lead to harmonicity conditions.
problem Understanding harmonicity of Bertrand curves.
method Developed new Frenet formulas and used them to write differential equations and harmonicity conditions.
result Sufficient conditions for harmonicity of Bertrand curves expressed in terms of the main curve.
We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…
The paper establishes gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
problem Gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
method Yau and Souplet-Zhang type gradient estimates for harmonic and heat equation solutions under Dirichlet boundary condition.
result Established gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
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.
This article has two purposes. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to Lie groups and symmetric spaces, focusing on spectral curves for harmonic 2-tori. The most unwieldy aspect of the spectral curve description is the periodicity conditions and th…
We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmon…
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.
Analyzes L2-harmonic forms on curved manifolds, proving integrability conditions.
problem Analyzing integrability of L2-harmonic forms on curved manifolds. method Established L∞-estimate via Moser iteration, proved vanishing of integrable forms. result Proves that L2-harmonic forms on non-positively curved manifolds are integrable if and only if they vanish. Maps converge to simpler structures under certain tension conditions.
problem Understanding convergence of maps under tension decay.
method Sharp criterion on tension decay ensures subconvergence to simpler structures.
result Maps subconverge to a structure made of harmonic maps.
Study pinched self-dual Weyl curvature in compact 4-manifolds.
problem Analyzing compact 4-manifolds with specific curvature properties.
method Examining harmonic self-dual Weyl curvature under pinching conditions.
result Characterized compact 4-manifolds with pinched self-dual Weyl curvature.
Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators …
In this note, using the spinorial description of SU(3) and G2-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique s…
Study Liouville theorem for specific harmonic maps.
problem Proving Liouville theorem for harmonic maps.
method Establish gradient estimates under specific conditions.
result Proved Liouville theorem for harmonic maps.
Gradient estimates for subelliptic harmonic maps with potential.
problem Estimating gradients of subelliptic harmonic maps.
method Investigation of subelliptic harmonic maps with potential from specific manifolds.
result Gradient estimates and Liouville type result established.
The concept of harmonic metallic structure on a metallic pseudo-Riemannian manifold is introduced. In the case of compact manifolds we prove that harmonicity of a metallic structure J, with J2=pJ+qI and p2+4q=0, is equivalent to dJ=0. Conditions for a harmonic metallic structure to be preserved by harmoni…
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. In this work, almost product and almost golden structures are studied. Conditions for those structures being Integrable and parallel are investigated. Also harmonicity of a map between almost pruduct or almost golden manifolds with pure or hyperbolic metric is discussed under certain conditions.
Study of quaternion-Kähler structures via harmonic flow on 8-manifolds.
problem Geometric flow of quaternion-Kähler structures on 8-manifolds.
method Formulated gradient Dirichlet flow, analyzed harmonicity, proved almost-monotonicity.
result Proved long-time existence and constructed harmonic solitons.
Conservation law for weakly harmonic mappings in high dimensions.
problem Conservation law for harmonic mappings in supercritical dimensions.
method Partial extension of Rivière's conservation law with Lorentz integrability condition.
result Conservation law for weakly harmonic mappings in supercritical dimensions.
Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal change…
Let p be a real number greater number greater than one. Suppose that a graph G of bounded degree is quasi-isometric with a Riemannian manifold M with certain properties. Under these conditions we will show that the p-harmonic boundary of G is homeomorphic to the p-harmonic boundary of M. We will also prov…
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.
Harmonic basis vector fields on surfaces
problem Parameterizing surfaces with harmonic vector fields
method Introducing harmonic basis vector fields and deriving conditions for their existence
result Classifying parameterizations of surfaces with harmonic basis vector fields
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.
The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves…
We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the a…