New classification of gradient steady Ricci solitons with vanishing D-tensor.
problem Classifying gradient steady Ricci solitons with specific properties.
method Extending Cao-Chen's work on Bach-flat gradient Ricci solitons, proving properties for D-flat solitons. result Any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat or isometric to the Bryant soliton. In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
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.
In this paper we prove that any n-dimensional (n≥4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant sol…
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.
The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.
problem The rigidity of positively curved gradient Ricci solitons on orbifolds.
method Analyzes scalar curvature, uses nonnegative curvature operator, κ-noncollapsed condition, and asymptotic quotient cylindrical properties.
result Steady gradient Ricci solitons on orbifolds with positive curvature are rigid and must be quotients of the Bryant soliton.
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.
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 soliton found on resolved cones.
problem Existence of Kähler-Ricci solitons on Calabi-Yau cones.
method Equivariant crepant resolutions and steady gradient Kähler-Ricci solitons.
result Unique complete steady gradient Kähler-Ricci soliton found.
New explicit Calabi-Yau metrics and Kähler-Ricci solitons found on complex n-space.
problem Constructing new explicit Calabi-Yau metrics and Kähler-Ricci solitons.
method Continuous (ℓ−1)-parameter family of explicit complete gradient steady Kähler-Ricci solitons on Cn with Hamiltonian 2-forms. result Construction of new complete gradient steady Kähler-Ricci solitons with positive sectional curvature.
The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton N2×R, the product of Hamilton's cigar soliton N2 and the real line R with the product metric. R. Hamilton has conjectured that there should exist a family of colla…
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.
New steady gradient Ricci solitons found on specific four-manifolds.
problem Finding steady gradient Ricci solitons on specific four-manifolds.
method Center manifolds and topological degree theory.
result New families of complete, SU(2)-invariant steady gradient Ricci solitons constructed. Paper studies fundamental groups of certain Ricci solitons.
problem Understanding fundamental groups of specific Ricci solitons.
method Analyzes properties of complete steady gradient Ricci solitons with nonnegative sectional curvature.
result Fundamental groups of these solitons are either trivial or infinite.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…
We show that for an n dimensional complete non Ricci flat gradient steady Ricci soliton with potential function f bounded above by a constant and curvature tensor Rm satisfying limr→∞r∣Rm∣<51, then ∣Rm∣≤Ce−r for some constant C>0, improving a result of [36]. For any f…
In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…
Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.
problem Characterize the geometry of steady gradient Ricci solitons at infinity.
method Analyze the rescaled limits of finite-time singular solutions of the Ricci flow.
result Classify the tangent flows at infinity of 4-dimensional steady soliton singularity models.
Researchers create a family of solitons connecting a cigar to a sphere.
problem Classifying steady gradient Ricci-Yang-Mills solitons on surfaces.
method Constructed a 1-parameter family of solitons on surfaces.
result Any complete steady gradient Ricci-Yang-Mills soliton on a surface must come from this family.
We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.
problem Characterizing gradient steady Kähler-Ricci solitons on toric manifolds.
method Analyzing orbit space structure and using Bakry-Émery tensor.
result Proves rigidity of solitons on complete toric manifolds.
New steady solitons found with SO(3) symmetry.
problem Finding steady solitons with specific symmetries.
method Proved existence of a family of solitons with SO(3) symmetry.
result Existence of a one-parameter family of solitons.
We show for a complete noncompact steady Ricci soliton that there exists a sequence {x_i} of points tending to infinity such that |Rc|(x_i) limits to zero.
The paper classifies invariant gradient k-Yamabe solitons in pseudo-Euclidean spaces.
problem Characterizing invariant gradient k-Yamabe solitons in pseudo-Euclidean spaces. method Characterization through the action of an (n−1)-dimensional translation group and classification of rotational invariant solutions. result Infinitely many explicit examples of geodesically complete steady gradient k-Yamabe solitons are constructed. We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.
Study of solitons in Laplacian flow on 7-manifolds.
problem Understanding finite-time singularities of Laplacian flow on G_2-structures.
method Systematic study of cohomogeneity-one solitons with Sp(2) and SU(3) symmetry groups.
result Existence and classification of smoothly-closing solitons, including complete and incomplete solutions.
We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove also a similar local splitting result under weaker curvature assumptions.
Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.
problem Investigate 4D gradient solitons with specific curvature properties.
method Analyze 4D gradient steady and shrinking solitons with nonnegative or half nonnegative isotropic curvature.
result Prove 2-nonnegativity of Ricci curvature and bound the curvature tensor for ancient solutions.
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.
New Kähler solitons found that are not U(n)-invariant.
problem Whether every steady gradient Kähler-Ricci soliton of positive curvature on Cn is U(n)-invariant. method Constructing a family of U(1)imesU(n−1)-invariant, but not U(n)-invariant, steady gradient Kähler-Ricci solitons. result Found a family of complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on Cn for n≥3. Researchers found cylindrical steady gradient solitons in 3D.
problem Finding steady gradient solitons in 3D with specific symmetries.
method Constructed a two-parameter family of solitons with SO(2)imesR symmetry. result Found a family of solitons with asymptotic power-law or exponential decay.
We produce non-Kähler complete steady gradient Ricci solitons generalising those constructed by Bryant and Ivey.
The purpose of this article is to study gradient Yamabe soliton on warped product manifolds. First, we prove triviality results in the case of noncompact base with limited warping function, and for compact base. In order to provide nontrivial examples, we consider the base conformal to a semi-Euclidean space, which is …
We show that Lorentzian manifolds whose isometry group is of dimension at least 21n(n−1)+1 are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally conformally flat and symmetric Lorentzian Ricci solitons which are not rigid.
We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…
New gradient Ricci solitons found via fiber bundles.
problem Existence of gradient Ricci solitons.
method Warped metrics and fiber bundles.
result Necessary and sufficient conditions for existence.
We show the existence and uniqueness of a one-parameter family of smooth complete U(1)-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano Kähler-Einstein base with first Chern class proportional to that of the base. These solitons are non-Kähler except on the total spac…
The article characterizes gradient ρ-Einstein solitons under specific conditions.
problem Characterizing gradient ρ-Einstein solitons with certain properties.
method Analyzing solitons with vector fields of bounded norm, finite weighted Dirichlet integral, and specific Ricci curvature restrictions.
result Non-trivial complete gradient ρ-Einstein solitons with finite weighted Dirichlet integral and certain Ricci curvature restrictions are of constant scalar curvature and steady.
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. Study on gradient Ricci solitons with isoparametric potential functions.
problem Characterizing gradient Ricci solitons with isoparametric potential functions.
method Analyzes complete gradient Ricci solitons with isoparametric potential functions, proving theorems for steady and shrinking cases.
result Critical level sets of codimension greater than one for steady case, partial results for shrinking case.
In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…
In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with no…
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 scalar curvature decay in four-dimensional steady solitons.
problem Behavior of scalar curvature at infinity on four-dimensional steady solitons.
method Analysis of scalar curvature decay rate and asymptotic cone properties.
result Linear scalar curvature decay away from edges, stronger inequality if scalar curvature vanishes at infinity.
3D steady gradient Ricci solitons are all O(2)-symmetric.
problem Characterizing 3D steady gradient Ricci solitons.
method Analyzing asymptotic behavior and using O(2) symmetry.
result All 3D steady gradient Ricci solitons are O(2)-symmetric.
By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtaine…
Paper proves volume growth estimate for steady gradient Ricci solitons.
problem Estimating the volume growth of steady gradient Ricci solitons.
method Proved a volume growth estimate using Nash entropy.
result Volume growth rate is no smaller than $r^{ rac{n+1}{2}}$.