The paper analyzes solitonic components of toric manifolds.
problem Analyzing solitonic components of toric manifolds.
method Computing eigenfunctions of a solitonic complex Laplacian operator.
result Determination of solitonic decomposition of Fano toric manifolds.
Study of Ricci-Yamabe solitons on Walker 3-manifolds.
problem Characterizing Ricci-Yamabe solitons on Walker 3-manifolds.
method Using Hodge decomposition of De-Rham, the soliton field is found from the potential function.
result Classification of all Ricci-Yamabe and gradient Ricci-Yamabe solitons in a Walker 3-manifold.
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.
This paper classifies Ricci solitons in complex hyperbolic spaces.
problem Understanding Ricci solitons in complex hyperbolic spaces.
method Analyzing homogeneous expanding Ricci solitons as submanifolds of complex hyperbolic spaces.
result Classification and analysis of Lie subgroups with Ricci soliton induced metric in complex hyperbolic spaces.
We extend to the Sasakian setting a result of Tian and Zhu about the decomposition of the Lie algebra of holomorphic vector fields on a Kähler manifold in the presence of a Kähler-Ricci soliton. Furthermore we apply known deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a stability result concern…
This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.
problem Classifying Ricci soliton subgroups in a specific type of nilpotent group.
method Using the properties of nilpotent Iwasawa groups and Lie subgroups.
result Classification of codimension one Lie subgroups of nilpotent Iwasawa groups that are Ricci solitons.
Study on G2-structures using Laplacian coflow and solitons.
problem Characterizing and understanding G2-structures and their solitons. method Using the irreducible G2-decomposition of the Hodge Laplacian and Lie derivative, characterizing infinitesimal symmetries and soliton conditions. result Proof of the absence of compact shrinking solitons for the Laplacian coflow.
We prove that a four-dimensional gradient shrinking Ricci soliton with δW±=0 is either Einstein, or a finite quotient of S3×R, S2×R2 or R4. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times\…
The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.
problem Decomposing the height function of different types of surfaces into simpler components.
method Using Euler-Ramanujan identities and Weierstrass-Enneper representation to decompose height functions of minimal, maximal, timelike minimal, and Born-Infeld surfaces.
result The height function of various surfaces can be expressed as a finite sum of scaled and translated versions of itself.
We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…
We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation o…
The Ahlfors Laplacian is applied to solve geometric and relativistic problems.
problem Solving geometric and relativistic problems using the Ahlfors Laplacian.
method Orthogonal decompositions and expansions of tensor components are used to study the Ahlfors Laplacian's applications.
result The Ahlfors Laplacian is applied to construct solutions of general relativistic constraint equations in vacuum.
We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the U-hierarchy of the ZS-AKN…
We propose an intuitive interpretation for nontrivial L2-Betti numbers of compact Riemann surfaces in terms of certain loops in embedded pairs of pants. This description uses twisted homology associated to the Hurewicz map of the surface, and it satisfies a sewing property with respect to a large class of pair-of-pa…
Characterizes limits of Ricci flows and their singularities.
problem Understanding the structure of non-collapsed limits of Ricci flows.
method Characterizes limits as smooth away from a set of high codimension, identifies tangent flows as gradient shrinking solitons, and stratifies singular set.
result Non-collapsed limits of Ricci flows are smooth away from a set of high codimension and have tangent flows as gradient shrinking solitons.
The paper classifies a type of solitons in Euclidean spaces.
problem Classifying generalized Yamabe solitons on hypersurfaces.
method Completely classified solitons arising from the position vector field.
result Classification of generalized Yamabe solitons on hypersurfaces in Euclidean spaces.
Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
New examples of solitons found as warped products.
problem Constructing new soliton examples.
method Warped products and explicit descriptions using elementary functions.
result Complete examples of Ricci almost solitons and Ricci-Bourguignon solitons.
Study on Yamabe solitons with applications and structure elucidation.
problem Understanding the structure of Yamabe solitons and their applications.
method Investigation of complete gradient conformal solitons under specific conditions.
result Affirmative partial answer to Yamabe soliton conjecture.
We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons. We classify Algebraic Ricci Solitons of three-d…
The study examines Riemann solitons and almost solitons on specific Kenmotsu manifolds.
problem Characterizing solitons on Kenmotsu manifolds.
method Analysis of Riemann solitons and gradient almost Riemann solitons on almost Kenmotsu manifolds.
result Construction of examples of Kenmotsu and (κ,μ)′-almost Kenmotsu manifolds. 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. Study on Ricci-like solitons and gradient solitons on specific manifolds.
problem Characterizing solitons on Sasaki-like almost contact B-metric manifolds.
method Introduced and studied Ricci-like solitons with arbitrary potential and gradient solitons. Proved properties of the Ricci tensor and soliton coefficients.
result Gradient almost Ricci-like solitons have constant soliton coefficients.
Study on geometric properties of second Ricci solitons.
problem Understanding the geometry of second Ricci solitons.
method Investigation of closed and compact second Ricci soliton manifolds, and immersed submanifolds as well as warped product manifolds.
result Investigation of geometric properties of second Ricci solitons.
The study classifies steady Ricci solitons based on geometric conditions.
problem Characterizing steady Ricci solitons under specific geometric constraints.
method Analyzing geometric conditions and applying them to classify solitons.
result Steady Ricci solitons are classified into specific types based on given conditions.
Study on p-biharmonic maps from gradient Ricci solitons, focusing on 2D cigar soliton.
problem Understanding p-biharmonic maps on gradient Ricci solitons.
method Analyzing p-biharmonic maps from gradient Ricci solitons, specifically 2D cigar soliton.
result Obtained results on p-biharmonic maps from gradient Ricci solitons, particularly on 2D cigar soliton.
The study characterizes spacetimes with specific solitons in f(R)-gravity.
problem Characterizing spacetimes with specific solitons in f(R)-gravity. method Analyzing η-Ricci solitons, gradient η-Ricci solitons, gradient Einstein Solitons, and gradient m-quasi Einstein solitons in perfect fluid spacetimes obeying f(R)-gravity. result Established conditions for the behavior of η-Ricci solitons and derived significant theorems about dark matter. We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon almost solitons and prove some results about them which generalize previous results for Ricci alm…
Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact q-solitons are stationary and q-flat. Paper shows constant σk-curvature for quasi k-Yamabe solitons.
problem Understanding constant curvature in quasi k-Yamabe solitons.
method Analyzes conditions for solitons to be gradient and constant curvature.
result Compact quasi k-Yamabe solitons have constant σk-curvature.
3D solitons classified into specific types.
problem Classifying 3D steady and expanding gradient Ricci solitons.
method Analyzing rectifiable potential functions and curvature conditions.
result 3D solitons are isometric to specific known forms.
Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.
problem Examining solitons on homogeneous manifolds to understand their properties and constraints.
method Analyzing ambient obstruction flow and specific solitons in homogeneous spaces, proving properties and constructing examples.
result Proved that any compact ambient obstruction soliton with constant scalar curvature is trivial, and characterized specific types of solitons in 4-dimensional homogeneous spaces.
The paper characterizes Ricci solitons on the Poincaré upper half plane.
problem Characterizing Ricci solitons on the Poincaré upper half plane.
method Classifying and generalizing Ricci solitons and soliton equations in the half plane of Poincaré.
result Obtained some properties of solitons about their geodesic flows.
Characterizes ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds.
problem Understanding ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds. method Analyzes the geometry of ∗-k-Ricci-Yamabe solitons and gradient solitons on Kenmotsu manifolds. result Characterizes the nature of ∗-k-Ricci-Yamabe solitons and gradient solitons. The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.
New families of Ricci solitons found with collapsing volume.
problem Finding new Ricci solitons with specific volume behavior.
method Reduced soliton equation to Monge-Ampère equation coupled with ODEs.
result Explicit complete expanding solitons and existence results for other types.
The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.
problem Classifying h-almost Ricci-Yamabe solitons in paracontact geometry.
method Characterization and classification of para-Kenmotsu, para-Sasakian, and para-cosymplectic manifolds.
result Characterizations and classifications of various paracontact manifolds.
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
Study on para-Sasakian metrics and their solitons.
problem Characterizing para-Sasakian metrics with conformal η-Ricci solitons.
method Analyzing the properties of para-Sasakian metrics under conformal η-Ricci solitons.
result Para-Sasakian metrics admitting conformal η-Ricci solitons are η-Einstein.
Study on steady solitons' curvature behavior near infinity.
problem Curvature estimate of steady solitons near infinity.
method Investigation of asymptotic curvature properties.
result Asymptotic curvature estimate for steady solitons.
The study classifies specific types of solitons with bounded scalar curvature.
problem Classifying quasi-Yamabe gradient solitons with bounded scalar curvature.
method Analyzing complete, nontrivial solitons with scalar curvature bounded above or below.
result Classification of specific types of solitons with bounded scalar curvature.
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
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.
We use the bracket flow/algebraic soliton approach to study the Laplacian flow of G2-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a G-invariant G2-structure on a homogeneous space G/K that flows by pull-ba…
The paper classifies 3D complete gradient Yamabe solitons.
problem Classifying nontrivial 3D complete gradient Yamabe solitons.
method Analyzing properties of Yamabe solitons to show rotationally symmetry.
result Nontrivial non-flat 3D complete steady gradient Yamabe solitons are rotationally symmetric.
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
In this paper, we consider ∗-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold M is a ∗-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a ∗-Ricci soliton, then we show that M is of constant sectional curvatu…