Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…
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). 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.
Study finds all 4D Lie groups with harmonic curvature.
problem Finding Lie groups with harmonic curvature in 4D.
method Analyzing pp-wave Lie groups and determining harmonic curvature conditions.
result Description of 4D pp-wave Lie groups obtained.
Rigidity proven for a specific type of solitons with harmonic curvature.
problem Proving rigidity of a specific class of solitons.
method Proof of rigidity for compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature.
result Compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature are rigid.
The study finds conditions for almost-Kähler 4-manifolds to be Kähler.
problem Conditions for almost-Kähler 4-manifolds to be Kähler.
method Analyzes harmonic self-dual Weyl curvature and constant scalar curvature.
result Compact almost-Kähler 4-manifolds with harmonic self-dual Weyl curvature and constant scalar curvature are Kähler if c1⋅ω≥0. 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 paper classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Extending the 4-dimensional work by Kim-Shin, the paper classifies n-dimensional spaces (n≥5). result New counterexamples to the Fischer-Marsden conjecture on compact vacuum static spaces.
Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
Paper solves Minkowski problem for p-harmonic measures.
problem Solving the Minkowski problem for p-harmonic measures on convex domains.
method Using the Gauss curvature flow method.
result Existence of smooth solution to the Minkowski problem for p-harmonic measures.
New findings on compact manifolds with specific curvature properties.
problem Characterizing compact Riemannian manifolds with harmonic Weyl curvature and curvature operator of the second kind.
method Analyzing the curvature properties and using the cone condition.
result Classification of manifolds with harmonic Weyl curvature and specific curvature operator properties.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
The paper estimates gradients and proves Liouville theorems for p-harmonic maps.
problem Estimating gradients and proving Liouville theorems for p-harmonic maps.
method Obtained an Lq gradient estimate for p-harmonic maps, derived from which a Liouville type result was obtained. result Established a gradient estimate and Liouville theorem for p-harmonic maps. The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.
problem The vanishing and finiteness of p-harmonic 1-forms on submanifolds.
method Using BiRic curvature conditions to prove theorems.
result Theorems on vanishing and finiteness of p-harmonic 1-forms.
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
Classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Thorough classification through geometric analysis.
result Spaces are locally isometric to four types.
Embedded minimal surfaces of finite total curvature in R3 are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in R3 of compact Riemann surfaces with finitely many punctures…
It was proved that the fundamental group of the space of harmonic polynomials of degree n(n≥2), with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic functions having the same Gaussian curvature.
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.
Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.
problem Investigating rigidity of specific gradient steady Ricci solitons.
method Proving rigidity for n-dimensional (n≥5) complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. result Proves that such solitons are either Ricci flat or isometric to the Bryant soliton up to scaling.
Localized curvature bounds ensure harmonic maps are constant.
problem Ensuring harmonic maps are constant under localized curvature constraints.
method Localized Bochner-type rigidity theorem for harmonic maps with image-dependent curvature bounds.
result Harmonic maps are constant if minimal Ricci curvature dominates image-dependent curvature bounds.
Study on harmonic functions on nonnegative curvature 3D manifolds.
problem Analyzing harmonic functions on specific 3D manifolds.
method Inspired by Miao, developed a monotonic quantity for level sets of harmonic functions on (R3∖{0},g) with nonnegative scalar curvature. result Established a rigidity result for the derived monotonic quantity.
Study on 3-manifolds with nonnegative scalar curvature and positive harmonic functions.
problem Characterizing 3-manifolds with nonnegative scalar curvature.
method Exhaustions by level sets of harmonic functions and refined average gradient estimates.
result Contractible 3-manifolds are diffeomorphic to R^3, and handlebodies have genus at most 1.
In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional c…
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.
The paper classifies quasi-Einstein manifolds with harmonic Weyl curvature.
problem Classifying quasi-Einstein manifolds with specific curvature properties.
method Extending and refining previous work on quasi-Einstein manifolds, focusing on harmonic Weyl curvature.
result New examples of quasi-Einstein manifolds are provided, which are neither locally conformally flat nor D-flat.
The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
Develops methods to construct harmonic and wave maps into variable-curvature surfaces.
problem Limited explicit constructions for harmonic and wave maps in variable-curvature settings.
method Reduction framework for pseudo-Riemannian surfaces, geometric ansatz, first-order ODEs.
result Constructs explicit harmonic and wave maps into ellipsoids, hyperboloids, and Schwarzschild exterior.
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
In this paper, we prove that the static triple with half harmonic Weyl curvature and positive scalar curvature must be the standard hemisphere.
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 …
Let Mn(n≥3) be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that Mn satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4,…
The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…
The paper proves Liouville theorems for V-harmonic maps under specific curvature conditions.
problem Proving Liouville theorems for V-harmonic maps in Riemannian manifolds with non-negative (m,V)-Ricci curvature. method Probabilistic proof extending previous results by Cheng, Hildebrandt-Jost-Wideman, and Stafford.
result Extends Liouville theorems to a broader class of manifolds and curvature conditions.
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.
Paper proves Liouville theorems for harmonic functions under specific curvature bounds.
problem Analyzing harmonic functions on manifolds with lower bounds of N-weighted Ricci curvature. method Uses Moser's iteration procedure to prove Liouville theorems.
result Establishes Liouville theorems for harmonic functions with sublinear growth and under weaker bounds of N-weighted Ricci curvature. 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.
One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.
problem Characterizing one-harmonic maps from curved surfaces to hyperbolic spaces.
method Using Minkowski geometry and interpreting maps as Gauss maps of convex surfaces.
result One-harmonic maps have images confined to the interior of convex hulls.
A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifold…
J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we describe the ordinary differential equations of 3-harmonic curves into a Riemannian manifold with constant sectional curvature, and show biharmonic curve is k-harmonic cu…
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
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.
In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature
We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
The study proves conditions for complete Riemannian manifolds to be Einstein.
problem Conditions for complete Riemannian manifolds to be Einstein.
method Proving conditions using harmonic curvature and curvature operator of the second kind.
result Complete Riemannian manifolds with specific curvature conditions are Einstein.
Paper extends foliation results in higher dimensions for Schwarzschild spaces.
problem Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds.
method Generalization to higher dimensions, proving existence under arbitrary dimensionality.
result Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds of arbitrary dimension.