The paper examines triviality of Ricci-Bourguignon harmonic solitons.
problem Investigating triviality of Ricci-Bourguignon harmonic solitons.
method Utilizing results from V-harmonic map to study Ricci harmonic soliton properties.
result Triviality of Ricci-Bourguignon harmonic solitons examined.
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.
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.
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.
This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci solit…
The concept of the Ricci soliton was introduced by Hamilton. Ricci soliton is defined by vector field and it's a natural generalization of Einstein metric. We have shown earlier that the vector field of Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an appl…
No nontrivial harmonic 1-forms on certain gradient Ricci solitons.
problem Existence of nontrivial harmonic 1-forms on gradient Ricci solitons.
method Proving Liouville theorems for harmonic 1-forms.
result No nontrivial L2-integrable harmonic 1-forms on specified solitons. 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 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.
Paper proves constant functions for pluriharmonic on certain solitons.
problem Proving Liouville type theorems for harmonic functions on gradient Ricci solitons.
method Analyzing pluriharmonic functions on gradient shrinking or steady Kähler-Ricci solitons.
result Any pluriharmonic function with gradient in Lp is constant. The paper classifies Ricci solitons and studies harmonic vector fields on a specific Thurston geometry.
problem Classifying Ricci solitons and studying harmonic vector fields in a specific Thurston geometry.
method Left-invariant Riemannian metric classification and analysis of harmonic maps and vector fields.
result All Ricci solitons on (F4,g) are expanding and non-gradient. The paper studies steady solitons with curvature decay and proves their smoothness.
problem Analyzing the properties of steady solitons with curvature decay.
method Bootstrap regularity in harmonic coordinates using the soliton equation.
result Steady gradient Ricci solitons are asymptotically cylindrical under certain curvature decay conditions.
In this paper, we study Riemannian submersions whose total manifolds admitting a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we st…
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.
The paper studies Riemannian maps with Ricci soliton base manifolds.
problem Analyzing Riemannian maps with specific properties of base manifolds.
method Analyzing Riemannian curvature tensor, Ricci tensor, scalar curvature, and necessary conditions for Ricci soliton leaves.
result Necessary and sufficient conditions for harmonicity and biharmonicity of Riemannian maps.
Study new Einstein-like metrics and their properties.
problem Characterize a new class of quasi-Einstein metrics.
method Investigate modified Ricci solitons and their relationships.
result Prove rigidity of standard spheres under specific conditions.
The paper is devoted to the study of the global geometries of harmonic mappings and infinitesimal harmonic transformations and presents their applications to the theory of Ricci solitons.
In the present paper, by using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as a generalization of recent works by H. Li, and M. Fernandez-…
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 on η-Ricci-Yamabe solitons on Riemannian submersions.
problem Characterizing η-Ricci-Yamabe solitons on Riemannian submersions. method Analyzing conditions for η-Ricci-Yamabe solitons on submersions and deriving Laplacian equations. result Classification of fiber and target manifolds as η-Ricci-Yamabe solitons under various conditions. 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…
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.
Classifies gradient Ricci solitons with harmonic Weyl curvature in dimensions 5 and above.
problem Classifying gradient Ricci solitons with harmonic Weyl curvature in higher dimensions.
method Developed a novel method of refined adapted frame fields and used geometric arguments.
result Local and complete classifications of gradient Ricci solitons with harmonic Weyl curvature.
In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic …
The study examines polynomial growth functions on gradient shrinking Ricci solitons.
problem Characterizing harmonic and caloric functions with polynomial growth on gradient shrinking Ricci solitons.
method Analysis of polynomial growth functions under different curvature conditions.
result Finite dimensional estimates for harmonic and caloric functions with polynomial growth.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
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 characterize η-Ricci solitons (g,ξ,λ,μ) in some special cases when the 1-form η, which is the g-dual of ξ, is a harmonic or a Schrödinger-Ricci harmonic form. We also provide necessary and sufficient conditions for η to be a solution of the Schrödinger-Ricci equation and point out the relation between …
In this paper we study gradient Ricci-Harmonic soliton with structure of warped product manifold. We obtain some triviality results for the potential function, warping function and the harmonic map which reaches maximum or minimum. In order to obtain nontrivial examples of warped product gradient Ricci-harmonic soliton…
The paper characterizes Clairaut conformal submersions on Ricci solitons.
problem Characterizing Clairaut conformal submersions on Ricci solitons.
method Calculating scalar and Ricci tensors, providing necessary conditions for fibres and base manifolds to be Ricci solitons and Einstein, and solving Poisson equations.
result Necessary and sufficient conditions for Clairaut conformal submersions to be harmonic.
The paper explores conformal submersions from Ricci solitons to Riemannian manifolds.
problem Characterizing conformal submersions from Ricci solitons.
method Analyzing properties of the O'Neill tensor, calculating Ricci tensors, and finding necessary conditions for harmonic maps.
result Necessary conditions for fibers and base manifold to be Ricci solitons, almost Ricci solitons, and Einstein.
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.
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319--327, 2010), Futaki and Sano (Asian J. Math. 17, 17-…
Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.
problem Understanding the long-time behavior of Ricci flows on manifolds.
method Prove equations dimension-reduce to twisted harmonic-Einstein equations, establish correspondence with G-Higgs bundles.
result Produce infinite families of new non-locally homogeneous examples, complete description in dimension 4.
We prove that there does not exist non-constant positive f-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the Lp(p≥1 or 0<p≤1) Liouville theorems on the complete gradient shrinking Ricci solitons.
In this paper we give some results on the topology of manifolds with ∞-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…
Study on solitons in specific geometric manifolds, proving manifold properties and presenting examples.
problem Exploring solitons in almost coKähler and asymptotically harmonic manifolds.
method Analyzing η-Einstein and gradient η-Einstein solitons in various geometric frameworks. result Proves manifold properties and presents examples validating results.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
New flow for G2-structures helps find torsion-free structures.
problem Finding torsion-free G2-structures on compact manifolds.
method Ricci-harmonic flow of G2-structures, analyzing Taylor series expansion.
result Stationary points of the flow are torsion-free G2-structures.
We study the soliton flow on the domain of a twistorial harmonic morphism between Riemannian manifolds of dimensions four and three. Assuming real-analyticity, we prove that, for the Gibbons-Hawking construction, any soliton flow is uniquely determined by its restriction to any local section of the corresponding harmon…
New stable shrinking Ricci soliton found in 4D.
problem Stability of 4D Ricci shrinkers.
method Explicit metric description, tensor harmonic analysis, Kähler structure, selfduality insights.
result First non-cylindrical linearly stable shrinking Ricci soliton in 4D.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…
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.
We obtain a formal obstruction, i.e. a necessary condition for the existence of polarised complex deformations of Kähler-Ricci solitons. This obstruction is expressed in terms of the harmonic part of the variation of the complex structure.
We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M,g,f) which have harmonic Weyl curvature, i.e. δW=0. Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \tim…
On an n-dimensional complete manifold M, consider an h-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and dh/du>0, then the manifold M is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. More…
The paper characterizes contact metric manifolds with specific solitons.
problem Characterizing contact metric manifolds with ∗-conformal Ricci solitons. method Analyzing properties of (2n+1)-dimensional N(k)-contact metric manifolds. result The manifold is locally isometric to a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4.