Study classifies compact 4-manifolds with specific curvature properties.
problem Classifying compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature.
method Analyzing properties of 4-manifolds to identify those with harmonic Weyl tensor and nonnegative biorthogonal curvature.
result Classification of compact 4-manifolds with the specified curvature properties.
Discretizes harmonic tensors on Riemannian manifolds.
problem Discretizing harmonic tensor fields on Riemannian manifolds.
method Reduces to scalarization for ordinary harmonic functions under certain holonomy restrictions.
result Lifted diffusion on orthonormal frame bundle has same recurrence property as original Brownian motion.
Study shows Sasaki solitons with harmonic Weyl tensor are spheres.
problem Characterizing gradient shrinking Sasaki-Ricci solitons.
method Integral curvature estimates and quotient analysis.
result Gradient shrinking Sasaki-Ricci solitons with harmonic Weyl tensor are finite quotients of spheres.
New proof shows gradient Ricci solitons with harmonic Weyl tensor have at most three eigenvalues.
problem Classifying gradient Ricci solitons with harmonic Weyl tensor.
method Shorter proof without moving frame, focusing on eigenvalues.
result Ricci tensor has at most three distinct eigenvalues.
Study vacuum static spaces with vanishing Bach and Weyl tensors, proving harmonicity and rigidity results.
problem Characterizing vacuum static spaces with specific tensor properties.
method Analyzing the complete divergence of Bach and Weyl tensors, proving conditions for harmonicity and rigidity.
result Proves the vanishing of complete divergence of Bach and Weyl tensors implies harmonicity of the metric.
Investigates Schouten-Weyl tensor on 3D Lie groups with specific metrics.
problem Analyzing the Schouten-Weyl tensor on 3D Lie groups with special metrics.
method Examines left-invariant Lorentzian metrics and investigates harmonicity of the tensor.
result Identifies specific Lie groups with zero Schouten-Weyl tensor.
Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.
problem Characterizing 4-manifolds with harmonic anti-self dual Weyl tensor.
method Proving isometry to geodesic balls in space forms.
result Simply connected critical metrics are geodesic balls in space forms.
Study classifies gradient almost Ricci solitons with harmonic Weyl tensor.
problem Characterizing the local structure of gradient almost Ricci solitons with harmonic Weyl tensor.
method Local representation as multiply warped products, analysis of eigenvalues, and classification based on Weyl tensor properties.
result Classification of gradient almost Ricci solitons with harmonic Weyl tensor, extending previous results.
In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n…
Study on pp-waves in isotropic quasi-Einstein manifolds.
problem Characterizing the local structure of quasi-Einstein manifolds.
method Analysis of Weyl tensor properties and quasi-Einstein equation.
result Isotropic quasi-Einstein manifolds with harmonic Weyl tensor are pp-waves. This paper classifies critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
problem Classifying critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
method Complete classification through geometric conditions and equivalence of assumptions.
result Critical metrics with harmonic Weyl tensor on simply connected compact manifolds with boundary are isometric to geodesic balls in simply connected space forms.
The paper proves Liouville-type theorems on Hadamard manifolds.
problem Non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds.
method Proofs use Liouville-type theorems on non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, modified for Hadamard manifolds.
result Proves several Liouville-type theorems on Hadamard manifolds.
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.
We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times t \in[0, T), then the curvature tensor has to be uniformly bounded as well.
Weak harmonic Weyl metrics found on all 4D closed manifolds.
problem Finding canonical metrics on 4D closed manifolds.
method Critical points of a quadratic functional involving the divergence of the Weyl tensor.
result Every 4D closed manifold admits a unique weak harmonic Weyl metric.
Study long-term behavior of Ricci-harmonic flow with curvature bounds.
problem Understanding long-term behavior of Ricci-harmonic flow with curvature constraints.
method Extended Cao's result for scalar curvature and Weyl tensor, generalized Simon's result for Riemann curvature.
result Established long-term existence and bounded scalar curvature in the Ricci-harmonic flow.
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.
Study finds rigidity results for manifolds with harmonic Weyl curvature.
problem Characterizing closed manifolds with harmonic Weyl curvature.
method Developed new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor.
result Generalized Tachibana's theorem for non-negative curvature operator.
The paper characterizes rigidity in harmonic-Ricci solitons.
problem Characterizing rigidity in harmonic-Ricci solitons.
method Introducing and characterizing rigidity for harmonic-Ricci solitons, providing characterizations and discussing different cases.
result Rigidity can be traced back to the vanishing of certain modified curvature tensors.
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.
Study rigidifies geometry of electrostatic systems with specific tensor properties.
problem Investigating rigidity in electrostatic systems with specific tensor properties.
method Analyzing static Einstein--Maxwell spacetimes with harmonic (anti-)self-dual Weyl tensor.
result Gradient of lapse function is an eigenvector of Ricci tensor and manifold is locally conformally flat.
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 p-harmonic coordinates, Hölder metrics lead to useful gauge conditions in conformal geometry.
problem Establishing useful gauge conditions for regularity in conformal geometry.
method Development of p-harmonic coordinates on Riemannian manifolds with Hölder continuous metrics.
result Conformal mappings between manifolds with Hölder metrics are C1+α regular. We prove harmonic maps into nonnegative curvature are totally geodesic.
problem Harmonic mappings into non-negatively curved manifolds.
method Developed a theory for harmonic mappings between compact manifolds with nonnegative sectional curvature.
result Any harmonic map between Riemannian manifolds with nonnegative sectional curvature is totally geodesic.
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.
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 …
Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.
problem Characterizing gradient ρ-Einstein solitons with specific tensor properties.
method Analyzing the properties of Bach tensor and using local warping to classify solitons.
result Gradient ρ-Einstein solitons with radially nonnegative Bach tensor are locally warped products of an interval and an Einstein manifold.
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.
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
The paper analyzes tensors in generalized Robertson-Walker space-times.
problem Analyzing tensors in generalized Robertson-Walker space-times.
method Proving theorems about Ricci and Weyl tensors, decomposing Ricci tensor, showing conditions for harmonic Weyl tensor, and generalizing Riemann tensor structure.
result Conditions for a GRW space-time to be a quasi-Einstein manifold and the structure of Riemann tensor.
The paper proves vanishing properties of p-harmonic ℓ-forms on various Riemannian manifolds.
problem Vanishing properties of p-harmonic ℓ-forms on Riemannian manifolds. method The approach involves studying complete non-compact immersed submanifolds, Riemannian manifolds with weighted Poincaré inequality, and complete simply connected, locally conformally flat Riemannian manifolds.
result The existence of nontrivial p-harmonic ℓ-forms is shown to vanish under certain geometric conditions. This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or 1-quasiregular mapping between two manifolds with Cr metric tensors (r>1) is a Cr+1 conformal (local) diffeomorphism. …
We study the biharmonic stress-energy tensor S2 of Gauss map. Adding few assumptions, the Gauss map with vanishing S2 would be harmonic.
The study classifies 4D spaces with harmonic curvature and related metrics.
problem Classifying 4D Riemannian manifolds with harmonic curvature.
method Analyzing solutions to a specific differential equation and using properties of the Ricci tensor.
result A neighborhood of any point in the manifold is locally isometric to one of five types of spaces.
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…
Paper proves conjecture about critical metrics with divergence-free Bach tensor.
problem Proving conjecture about critical metrics with specific curvature properties.
method Used divergence-free Bach tensor to prove conjecture.
result Proved conjecture about critical metrics with divergence-free Bach tensor.
In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a war…
We descrive examples of metrics in the conformal class [g] on complete conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with non parallel Ricci tensor.
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.
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. New rigidity results for tensors on non-compact manifolds with curvature conditions.
problem Rigidity phenomena for tensors on non-compact Riemannian manifolds.
method Extending Bochner technique to non-compact settings, using Lichnerowicz Laplacian.
result Vanishing and rigidity of curvature tensors on Ricci-flat and Einstein manifolds.
New algorithms learn multi-index models via harmonic analysis, achieving statistical and computational trade-offs.
problem Learning multi-index models with unknown projections of input data.
method Exploiting the equivariance of the problem under the orthogonal group, we derive lower bounds and construct spectral algorithms based on harmonic tensor unfolding.
result Achieve statistical and computational trade-offs between sample and runtime complexity.
In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.
The paper explores geometric decompositions for Ricci tensors and their applications.
problem Understanding Ricci tensors on compact Riemannian manifolds.
method Utilizes Berger-Ebin and York L2-orthogonal decompositions. result New insights into Ricci almost solitons and harmonic maps.
Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact a…
Proves harmonic coordinates for weak immersions in even dimensions.
problem Existence of harmonic coordinates for weak immersions in Sobolev spaces.
method Analyzes weak immersions in critical Sobolev spaces and uses smallness conditions on the second fundamental form.
result Global harmonic coordinates exist for weak immersions in even dimensions under certain conditions.
The paper proves rigidity of certain Ricci solitons with specific curvature conditions.
problem Proving rigidity of gradient shrinking Ricci solitons with weakly harmonic Weyl tensors.
method Analyzing the Weyl curvature tensor and its relation to Ricci solitons.
result Gradient shrinking Ricci solitons with weakly harmonic Weyl tensors are Einstein.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.