The paper defines Hesse solitons and explores their properties on Hessian manifolds.
problem Exploring self-similar solutions to the Hesse flow on Hessian manifolds.
method Defining Hesse solitons and analyzing their properties on Hessian manifolds.
result Compact proper Hesse solitons are expanding, and non-trivial compact gradient Hesse solitons are proper.
In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …
We prove that any shrinking Kahler Ricci soliton has only one end, and that any expanding Kahler Ricci soliton with proper potential has only one end.
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.
It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or a product R×N(c), where N(c) is a space of constant curvature.
In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic inva…
Study of η-Ricci solitons on Kenmotsu 3-manifolds.
problem Exploring η-Ricci solitons on Kenmotsu 3-manifolds. method Examined various types of η-Ricci solitons on Kenmotsu 3-manifolds, including those with specific curvature conditions. result Existence of proper η-Ricci solitons on Kenmotsu 3-manifolds. 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.
The paper establishes curvature estimates for solitons in higher dimensions.
problem Curvature estimates for steady and expanding solitons in higher dimensions.
method Curvature estimates using gradient Ricci solitons and integral estimates.
result Curvature operator decays at specific rates for different cases of solitons.
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.
Kähler soliton surfaces are typically toric under generic conditions.
problem Understanding the symmetry of Kähler gradient Ricci solitons.
method Analyzing the integrability of the Hamiltonian system and the toric action.
result Kähler soliton surfaces are generically toric under certain conditions.
We study some potential theoretic properties of homothetic solitons Σn of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in Rn+m, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that par…
In this paper, we introduce a class of Sasaki manifolds with a reductive G-group action, called G-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessar…
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.
In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for f-stability index of a constant weighted mean curvature hypersurface…
Let (X,ω) be a compact normal Kähler space, with Hodge metric ω. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on Hω(X), the space of Kähler potentials, whose completion is the fin…
Abstract: Generalizes stability theories over toric varieties and Novikov type rings.
problem Stability theories over toric varieties and Novikov type base.
method Generalization of stability theories to families over toric varieties and their analytic analogues.
result Established properness of moduli of Calabi-Yau cones and Kahler-Ricci solitons.
We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…
In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified K-energy is proper f…
New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.
problem Mean curvature flow in infinite volumes with finite energy.
method Introducing calibration energy and proving its dissipation identity for mean curvature flows.
result Every proper self-expander with finite calibration energy is a plane in all dimensions and codimensions.
Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the origin…
Let M be a compact complex manifold admitting a Kähler structure. A conformally Kähler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric g~ on M with constant scalar curvature such that there is a positive smooth function f with g=f2g~ being a Kähler metric and f being…
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 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.
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.
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.
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.