Researchers prove isoperimetric inequality in specific steady Ricci solitons.
problem Proving the isoperimetric inequality in steady Ricci solitons.
method Utilized Guan-Li-Wang's result on warped product metrics and analyzed the soliton structure.
result Proved the isoperimetric inequality in the cigar and Bryant steady solitons.
Study on noncompact steady quasi-Einstein manifolds with specific tensor conditions.
problem Classifying noncompact steady quasi-Einstein manifolds with vanishing Weyl tensor condition.
method Analyzing manifolds with nonnegative Ricci curvature and zero radial Weyl curvature under fourth-order divergence-free Weyl tensor condition.
result Proves that such manifolds must be a warped product with (n−1)−dimensional Einstein fiber. New steady Euler flows found on 3-sphere and Sasakian manifolds.
problem Finding new steady Euler solutions on specific manifolds.
method Bifurcating from an existing ansatz and extending it to Sasakian 3-manifolds.
result Previously known solutions are not isolated, and new solutions found on 3-sphere and other Sasakian manifolds.
Study 4D steady gradient Ricci solitons reducing to 3D manifolds.
problem Understanding 4D steady gradient Ricci solitons that reduce to 3D.
method Analyzing asymptotic geometry and curvature properties.
result 4D solitons either reduce to spherical space forms or the 3D Bryant soliton.
Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
problem Classifying steady Euler flows with Morse-Bott Bernoulli functions.
method Constructing non-vanishing steady solutions using integrable systems and topology.
result Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph three-manifolds.
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 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. The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold (Mn,g) with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstei…
Characterizes 3D steady Euler flows using homologies.
problem Characterizing 3D steady Euler flows.
method Using commuting zero-flux homologies.
result Steady Euler flows cannot be constructed using plugs.
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.
New decay estimates for scalar curvature of steady gradient Ricci solitons.
problem Understanding scalar curvature behavior in steady gradient Ricci solitons.
method Using μ-bubbles introduced by Gromov.
result Provide new decay estimates for scalar curvatures.
Introduces a new length functional for Ricci flow to detect steady solitons.
problem Detecting steady solitons in Ricci flow.
method Develops a modified length functional and shows it satisfies differential inequalities.
result The length functional generates a distance function that saturates on steady soliton manifolds.
Study gradient Yamabe solitons on warped product manifolds.
problem Characterize and provide examples of gradient Yamabe solitons on warped product manifolds.
method Prove triviality results and provide nontrivial examples using a conformal base and translation group action.
result Provide infinitely many explicit examples of complete steady gradient Yamabe solitons.
New steady Ricci solitons found in even dimensions, including a four-dimensional example.
problem Existence of non-collapsed steady Ricci solitons in even dimensions.
method Construction of a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons on complex line bundles over Kähler-Einstein manifolds.
result Existence of new non-collapsed steady Ricci solitons in even dimensions, including a four-dimensional example.
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.
Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
problem Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
method Using open books, proved existence of non-vanishing steady solutions to the Euler equations for vector fields in odd dimensions.
result Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
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…
The paper proves convergence of normalized Ricci flow on compact manifolds.
problem Convergence of normalized Ricci flow on compact manifolds.
method Gradient inequality of Łojasiewicz type to show convergence to steady-states.
result Convergence of normalized Ricci flow to steady-states on compact manifolds.
New steady Kähler-Ricci solitons found on orbifolds.
problem Constructing steady Kähler-Ricci solitons on orbifolds.
method Using asymptotically cylindrical structures and orbifold actions.
result Found new examples of gradient steady Kähler-Ricci solitons converging to a Ricci-flat cylinder.
Paper proves compact manifolds can't have certain metrics.
problem Compact Riemannian manifolds and metrics with positive scalar curvature.
method Analyzes functions and geodesic rays in compact manifolds.
result Compact manifolds cannot have metrics with positive scalar curvature.
Study on gradient steady Kähler Ricci solitons with specific curvature properties.
problem Characterize gradient steady Kähler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature.
method Analyzing the structure of these solitons and their universal covering spaces.
result Gradient steady Kähler Ricci solitons are quotients of specific manifolds.
A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consi…
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…
Ancient flows of elliptic functionals classified in various dimensions.
problem Classifying ancient solutions to gradient flows of elliptic functionals.
method Analyzing closed ancient solutions in Riemannian manifolds.
result Ancient solutions classified in multiple dimensions and cases.
Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.
problem Solving CR Poincaré-Lelong equation on CR manifolds.
method Solves CR Poisson equation on CR (2n+1)-manifolds with specific curvature properties. result Discovers structures and CR Yamabe steady solitons on complete noncompact Sasakian manifolds.
Paper constructs complete Ricci solitons using growth estimates.
problem Constructing complete Ricci solitons in specific geometric setups.
method Growth estimate on soliton potential for cohomogeneity one manifolds, used to build families of solitons.
result Continuous families of complete steady and expanding Ricci solitons constructed.
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.
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.
Paper studies stability of generalized Ricci solitons with mathematical rigor.
problem Stability of generalized Ricci solitons under curvature assumptions.
method Computed second variation formula, proved stability conditions.
result Equivalence of dynamical and linear stability on steady gradient solitons.
New techniques analyze steady fluid flows on non-compact manifolds.
problem Analyzing steady fluid flows on non-compact manifolds with cylindrical ends.
method New techniques using b-calculus and b-symplectic structures. result New proofs and descriptions of b-symplectic structures on singular sets of the Bernoulli function. 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.
Characterizes potential function of almost conformal Ricci solitons on Sasakian manifolds.
problem Characterizing potential functions of almost conformal Ricci solitons on Sasakian manifolds.
method Characterization through the potential function f and non-dynamical scalar field p. result Established a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton.
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…
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
The paper studies Cotton solitons on specific geometric manifolds.
problem Analyzing Cotton solitons in almost Kenmotsu 3-h-manifolds. method Examined potential vector fields and their relationship with the Reeb vector field.
result Steady Cotton solitons on non-Kenmotsu manifolds are locally isometric to H2(−4)imesR. The paper classifies certain types of Ricci solitons with specific curvature properties.
problem Classifying gradient steady Ricci solitons with linear curvature decay.
method Analyzing the curvature properties and using classification techniques.
result Classification of 3D and 4D steady Ricci solitons with nonnegative curvature under linear decay.
Compact support steady flow in 3D Euler equations found.
problem Finding a nontrivial steady Euler flow with compact support.
method Constructing a nontrivial smooth steady incompressible Euler flow in three dimensions with compact support.
result A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support constructed.
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.
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.
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.
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 $r^{ rac{n+1}{2}}$.
This research shows that steady solitons in higher dimensions always reduce at infinity.
problem Characterizing steady solitons with nonnegative sectional curvature in higher dimensions.
method Dimension reduction analysis and tangent flow classification.
result Steady solitons in higher dimensions always reduce at infinity.
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. The study examines quasi-Einstein structures on specific types of manifolds.
problem Characterizing quasi-Einstein structures on almost cosymplectic manifolds.
method Analyzes properties of quasi-Einstein structures on almost cosymplectic manifolds, Lie groups, and K-cosymplectic manifolds.
result Closed quasi-Einstein structures on compact almost cosymplectic manifolds do not exist.
Formula adjusts steady-state models for control confounding.
problem Learning steady-state models from operational data can be flawed due to control confounding.
method Derives a formula to adjust for control confounding using structural dynamical causal models.
result Estimates a causal steady-state model from closed-loop operational data.
In this paper, we analyze the asymptotic behavior of κ-noncollapsed and positively curved steady Ricci solitons and prove that any n-dimensional κ-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.
Method learns CTMC models from steady-state data, predicting unseen states.
problem Learning CTMC models from aggregate steady-state statistics without sequence examples.
method ∞-SGD, a stochastic gradient descent method that avoids infinite sums.
result Successfully learns CTMC models and predicts unseen states.