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.
Study expanding solitons on complex Lie groups with specific algebraic structures.
problem Investigate expanding solitons on complex Lie groups with left-invariant metrics.
method Analyze the algebraic structure of complex Lie groups and their decompositions.
result Show that the Lie algebras of complex Lie groups decompose into semidirect products.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Sharp curvature estimates for expanding Ricci solitons in various dimensions.
problem Estimating curvature bounds for expanding Ricci solitons.
method Sharp lower and upper bounds derived for scalar curvature under specific conditions.
result Sharp curvature estimates provided for expanding Ricci solitons in dimensions three and four.
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.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
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.
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.
New example of non-Kähler soliton with Kähler-like behavior at infinity.
problem Constructing non-Kähler expanding gradient Ricci solitons.
method Asymptotically conical (AC) construction with Kähler tangent cone at infinity.
result Example of a non-Kähler soliton with a Kähler-like behavior at infinity.
The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
Constructs expanding gradient Ricci solitons with unique properties.
problem Creating expanding gradient Ricci solitons with specific characteristics.
method Combining previous work with localized maximum principle.
result Constructs various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay.
Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.
problem Estimating volume growth for gradient Ricci solitons.
method Survey and prove new volume growth estimates.
result New volume growth estimates for expanding gradient Ricci solitons.
The paper pinches curvature in expanding Ricci solitons.
problem Curvature pinching in expanding Ricci solitons.
method Hamilton-Ivey type curvature pinching estimates.
result Three-dimensional Hamilton-Ivey type curvature pinching theorem.
The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in R3. result Translating and self-expanding solitons have finite topology under certain conditions.
We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem concerning nonnegatively curved expanding gradient Ricci solitons.
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.
Unique continuation result for expanding Ricci solitons.
problem Unique continuation of expanding Ricci solitons.
method Optimal relative integral convergence rate, relative entropy.
result Well-defined relative entropy for expanding solitons.
We show that expanding Kähler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to Kähler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.
Study expanding gradient Ricci solitons with Euclidean base.
problem Characterize expanding gradient Ricci solitons with specific properties.
method Analyze warped products with Euclidean base and invariant warping functions.
result Derive complete examples of expanding gradient Ricci solitons.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…
New solitons found for G2-Laplacian flow on Lie groups.
problem Existence of solitons for G2-Laplacian flow. method Existence proof for expanding and steady solitons.
result First example of non-extremally Ricci pinched steady soliton.
This paper classifies all expanding Ricci solitons on surfaces.
problem Identifying all expanding Ricci solitons on surfaces.
method Developed a Ricci flow existence theory and used uniqueness theory to classify solitons.
result Classified all expanding Ricci solitons on surfaces.
We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.
We produce new examples of non-Kähler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature.
In this paper, we study gradient Ricci expanding solitons (X,g) satisfying Rc=cg+D2f, where Rc is the Ricci curvature, c<0 is a constant, and D2f is the Hessian of the potential function f on X. We show that for a gradient expanding soliton (X,g) with non-negative Ricci curvature, the scalar curva…
We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.
In this paper, we prove that expanding gradient Ricci solitons with (positively) pinched Ricci curvature are trivial ones. Namely, they are either compact or flat.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…
Study shows expanding Ricci solitons from specific metric cones.
problem Analyzing Ricci flows from weakly PIC1 metric cones.
method Complete weakly PIC1 Ricci flows with Euclidean volume growth.
result Ricci flows must be expanding gradient Ricci solitons.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.
problem Characterizing the ends of complete gradient Schouten solitons.
method Analysis of ends without additional assumptions, focusing on shrinking and expanding cases.
result Finitely many ends for shrinking Schouten solitons and connected infinity for expanding ones.
We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…
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…
We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R2×M2×⋯×Mr, where r≥2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…
The study provides estimates for curvature of gradient Ricci solitons.
problem Estimating curvature of gradient Ricci solitons.
method Hamilton-Ivey estimates applied to steady and expanding gradient solitons.
result Quantitative lower bounds on curvature for specific solitons.
The paper classifies solitons for a specific type of flow.
problem Classifying solitons for a fully non-linear Yamabe flow.
method Careful analysis of an associated dynamical system.
result Existence and description of solitons for certain dimensions.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.
problem Understanding the behavior of gradient expanding Ricci solitons with finite scalar curvature ratio.
method Analyzing complete gradient expanding Ricci solitons with nonnegative Ricci curvature.
result Riemann curvature tensor must have at least sub-quadratic decay for finite asymptotic scalar curvature ratio.
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.
The study finds a lower bound for the diameter of gradient ρ-Einstein solitons.
problem Estimating the diameter of gradient ρ-Einstein solitons.
method Using mathematical conditions and properties of solitons to derive a lower bound.
result A lower bound for the diameter of gradient ρ-Einstein solitons is established.
We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any n∈N0 and…
We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.
We study the asymptotic volume ratio of non-steady gradient Ricci solitons. Moreover, a local estimate of the volume ratio is obtained for expanding solitons which satisfy limdist(O,x)→∞∣Sect∣⋅dist(O,x)2=0. Therefore, for such a soliton, we can show that it must have Rn as one of…
We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.
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…