The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
Study shows Sasaki solitons with harmonic Weyl tensor are spheres.
problem Characterizing gradient shrinking Sasaki-Ricci solitons.
method Integral curvature estimates and quotient analysis.
result Gradient shrinking Sasaki-Ricci solitons with harmonic Weyl tensor are finite quotients of spheres.
The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.
problem Proving the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
method Deriving uniform L^{4}-bounds and analyzing the conic Sasaki-Ricci flow.
result Existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.
problem Characterizing Sasaki-Ricci solitons on Sasakian manifolds of up to seven dimensions.
method Analysis of the Sasaki-Ricci flow and convergence to solitons.
result Existence and classification of Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.
The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.
problem Understanding the geometry and topology of Sasaki-Ricci solitons.
method Analyzing the properties of complete gradient shrinking Sasaki-Ricci solitons, proving connectedness at infinity and compactness under certain curvature conditions.
result Proves that Sasaki-Ricci solitons are either connected at infinity or compact, generalizing results from previous studies.
Gradient almost para-Ricci-like solitons have constant coefficients and scalar curvatures.
problem Characterizing gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds. method Proving constant coefficients and scalar curvatures through analysis of soliton properties.
result Constant coefficients and scalar curvatures for gradient almost para-Ricci-like solitons.
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…
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.
We show that on a Sasakian 3-sphere the Sasaki-Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki- Ricci soliton. We also show the existence and uniqueness of gradient Sasaki-Ricci soliton on each Sasakian 3-sphere.
Local Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds.
problem Characterizing local immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds.
method Analyzing local Sasakian immersions of Sasaki-Ricci solitons into fiber products of homogeneous Sasakian manifolds.
result Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds. Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.
problem Characterizing log del Pezzo surfaces with specific geometric properties.
method Examining two classes of non-toric log del Pezzo surfaces and analyzing their geometric properties.
result Examples found that admit Kähler-Ricci solitons but not Sasaki-Einstein cone links.
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…
Study of para-Ricci-like solitons on specific Riemannian manifolds.
problem Characterizing para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds. method Introduced and studied para-Ricci-like solitons with arbitrary potential. Proved properties of Ricci tensor and scalar curvatures.
result Ricci tensor is a constant multiple of the vertical component of both metrics, leading to equal and constant scalar curvatures.
Sasakian immersions prove Sasaki-Ricci solitons are η-Einstein with rational constants.
problem Understanding local immersions of Sasaki-Ricci solitons into Sasakian space forms.
method Analyzing local Sasakian immersions and proving η-Einstein properties.
result Sasaki-Ricci solitons are η-Einstein with rational constants under certain conditions.
Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström and coupled Kähler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields…
Yamabe solitons defined on specific Sasaki-like manifolds.
problem Defining Yamabe solitons on Sasaki-like almost contact B-metric manifolds.
method Contact conformal transformation of manifold components to define Yamabe solitons.
result Explicit 5-dimensional Lie group example of a Yamabe soliton.
Study para-Ricci-like solitons on special Riemannian manifolds, proving geometric properties and providing an example.
problem Characterize para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds. method Analyzed different cases of potential vectors and proved geometric properties of constructed objects.
result Obtained results for a parallel symmetric second-order covariant tensor and provided an explicit example.
New solitons defined for Sasaki-like almost contact complex Riemannian manifolds.
problem Characterizing new solitons in Sasaki-like almost contact complex Riemannian manifolds.
method Defined β-Ricci-Bourguignon-like almost solitons with special potential. result Characterized geometrically and constructed examples of new solitons.
The paper studies para-Sasaki-like manifolds with a new metric connection.
problem Investigating new geometric structures on para-Sasaki-like manifolds.
method Deriving relations between connections, analyzing curvature tensors, studying solitons, constructing examples.
result Derived relations and properties of para-Sasaki-like manifolds with the generalized symmetric metric connection.
We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the resul…
Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.
problem Hamilton-Tian conjecture for specific Sasakian manifolds.
method Sasaki-Ricci flow, compact transverse Fano Sasakian 5-manifolds, klt foliation singularities.
result Confirmed Hamilton-Tian conjecture for compact transverse Fano Sasakian 5-manifolds.
Study on Ricci-like solitons on specific geometric manifolds, finding properties and conditions.
problem Characterizing Ricci-like solitons on Sasaki-like almost contact B-metric manifolds.
method Analyzing cases with specific potential fields and studying curvature conditions.
result Found conditions for the potential to have constant length and manifold to be η-Einstein. Study on Ricci-like solitons on specific geometric manifolds.
problem Characterizing Ricci-like solitons on almost contact B-metric manifolds.
method Introduced and analyzed Ricci-like solitons with Reeb vector fields on these manifolds, considering special cases and providing examples.
result Ricci-like solitons on these manifolds coincide with Einstein-like structures.
We classify simply connected compact Sasaki manifolds of dimension 2n+1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…
In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f1 for the first…
The paper extends metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.
problem Extension of metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.
method Verification of the extension of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space of positive rational powers of the canonical line bundle.
result The extended metric along the zero section has an expression that can be extended to the total space, and restricts to a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to g-solitons on quasi-regular quotients. By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from Kähler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we e…
We describe a general procedure for constructing new Sasaki metrics of constant scalar curvature from old ones. Explicitly, we begin with a regular Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact manifold M and construct a sequence, depending on four integer parameters, of rays of constant scal…
This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.
problem Existence and properties of weighted constant scalar curvature Kähler metrics.
method Introducing a weight function g(v,w) and proving equivalence between (v,w)-CSCK metrics and g(v,w)-solitons. result Existence of (v,w)-CSCK metrics in the first Chern class is equivalent to existence of g(v,w)-solitons. Study of Yamabe solitons on specific geometric manifolds.
problem Characterizing Yamabe solitons on almost contact complex Riemannian manifolds.
method Investigation of two cases: Sasaki-like and torse-forming potentials.
result Explicit examples and theoretical properties confirmed in 3D.
We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform C1 bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…
The paper studies how certain solitons on Fano manifolds extend to nearby deformations.
problem Conditions for weighted solitons to extend to nearby deformations of Fano manifolds.
method Analyzes the Kuranishi family of Fano manifolds and uses equivariant automorphism groups.
result All members of the Kuranishi family of a Fano manifold with a weighted soliton have weighted solitons if and only if the dimensions of their T-equivariant automorphism groups are equal to that of the original manifold.
For the Riemannian manifold Mn two special connections on the sum of the tangent bundle TMn and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space Mn has a constant sectional curvature ±1. The geometric explanation of this property is given. This …
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
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…
Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.
problem Characterizing extremal Kahler and Sasaki metrics using energy coercivity.
method Maximal complex torus, coercive weighted Mabuchi energy, K-polystability.
result Coercive weighted Mabuchi energy implies strict positivity of Donaldson-Futaki invariant and existence of extremal metrics.
Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.
problem Extending classical extremal Kähler metrics to include new objects.
method Surveying recent works on weighted extremal Kähler metrics and the YTD conjecture.
result Survey of recent research on weighted extremal Kähler metrics.
Exploring Sasaki metrics in joined manifolds.
problem Existence of constant scalar curvature Sasaki metrics in joined manifolds.
method Investigating the Sasaki cone of joined manifolds and considering continuous families of extremal Sasaki twins.
result Conditions for the existence of constant scalar curvature Sasaki metrics in joined manifolds.
Negative curvature proven in Sasaki manifold space completion.
problem Curvature of Sasaki manifold completion.
method Mabuchi metric on Sasaki potentials space.
result Metric completion negatively curved in Alexandrov sense.
Blowups of Kähler manifolds can inherit extremal metrics.
problem Extending extremal metrics to blowups of Kähler manifolds.
method Analyzing the action of a torus on blowups and weighted extremal metrics.
result Blowups of Kähler manifolds can inherit weighted extremal metrics.
There is an obstruction to the existence of Kähler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler -Ricci solitons, Sasaki-Einstein metrics and (con…
Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagran…
We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relati…
A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…
New curvature for weighted Sasaki sphere found.
problem Classifying Sasaki manifolds.
method Using shifted cones introduced by Yang and Zhang.
result New curvature characterization for weighted Sasaki sphere.
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.
problem Existence and non-existence of constant scalar curvature and extremal Sasaki metrics.
method Analyzing natural Sasaki-Boothby-Wang manifolds and extremal Sasaki metrics on admissible projective bundles.
result The extremal Sasaki--Reeb cone is not necessarily connected and can be empty even in the non-Gorenstein case.