The paper examines conditions for conformal Ricci solitons on generalized (κ,μ)-space forms.
problem Conditions for conformal Ricci solitons on generalized (κ,μ)-space forms. method Derivation of conditions for solitons to be shrinking, steady, or expanding in terms of conformal pressure p.
result Conditions for a Ricci semi-symmetric generalized (κ,μ)-space form to form an Einstein manifold when equipped with a conformal Ricci soliton. The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.
problem Characterizing solitons and their dual forms.
method Necessary and sufficient conditions for the dual form to be harmonic or Ricci harmonic are derived.
result Conditions for the dual form of a soliton to be harmonic or Ricci harmonic are provided.
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. Hyperbolic solitons on trans-Sasakian space forms and their submanifolds
problem Characterizing hyperbolic solitons on trans-Sasakian space forms and their submanifolds
method Introducing hyperbolic ∗−Ricci solitons and hyperbolic Ricci-Yamabe solitons result Characterizing the nature of hyperbolic solitons on trans-Sasakian space forms and their submanifolds
The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.
problem Chen-Ricci inequalities for Riemannian submersions and maps.
method General forms of Chen-Ricci inequalities for Riemannian submersions and maps are derived, involving curvatures of subspaces.
result New, easy, and elegant techniques for Chen-Ricci inequalities are established.
This note analyzes the normal form of gradient Ricci 4-solitons.
problem Understanding the curvature operator of gradient Ricci 4-solitons.
method Analyzing the normal form of the operator R^+21H^ and curvature operator R^ of Koiso-Cao soliton. result The curvature operator of the Koiso-Cao soliton inherits a normal form relative to the space of algebraic Kähler curvature operators.
Hypersurfaces with constant Ricci eigenvalues in real space forms are classified.
problem Classification of curvature homogeneous hypersurfaces in real space forms
method Proving the converse of curvature homogeneity implies constant Ricci eigenvalues
result Hypersurfaces with constant Ricci eigenvalues in real space forms are classified
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.
The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.
problem Conditions for scalar-flat Kähler surfaces with special tensor properties.
method Conjecture and prove in three special cases.
result The conjecture is proven in three special cases.
In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field ξare discribed and Ricci soliton in perfect fluid spacetime with torse-forming vector field ξare determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.
Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.
problem Characterizing Kähler-Ricci solitons that can be immersed into complex space forms.
method Proving that a Kähler-Ricci soliton's metric is Einstein if it can be immersed into a complex space form.
result The Kähler-Ricci soliton's metric is an Einstein metric if it can be immersed into a complex space form.
Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of ∗-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a ∗-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex …
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
The paper sets new bounds for Ricci-curvature in submersions and maps.
problem Establishing bounds for Ricci-curvature in submersions and maps.
method Developed upper and lower bounds for Ricci-curvature in submersions and maps, providing geometric characterizations.
result New bounds for Ricci-curvature in submersions and maps have been derived.
In this paper, we study the evolution of L2 one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L2 norm of a smooth one form with compact support is non-increasing along the Ricci flow with bounded curvature. The L∞ norm is showed to…
The paper characterizes ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds.
problem Characterizing ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. method Analyzing conditions for compressing, balancing, or enlarging ∗-Ricci-Bourguignon on Kenmotsu manifolds; estimating curvature properties; featuring with torse-forming vector fields; providing an example. result Found conditions and curvature properties for ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose ∗-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel ∗-Ricci tensor in case…
In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ric…
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
Study on gradient pseudo-Ricci solitons on real hypersurfaces.
problem Characterize gradient pseudo-Ricci solitons on real hypersurfaces.
method Analyze real hypersurfaces in complex space forms with specific eigen properties of the Ricci tensor.
result Show existence of non-trivial gradient pseudo-Ricci solitons on 3D ruled real hypersurfaces.
The paper derives curvature inequalities for submersions from quaternionic space forms.
problem Deriving curvature inequalities for submersions from quaternionic space forms.
method Analyzing Ricci and scalar curvatures of horizontal and vertical distributions in anti-invariant submersions.
result Established Ricci curvature inequality for anti-invariant submersions.
In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a smooth p-form is non-increasing along the Ricci flow. The L^{\infty} norm is showe…
Surveying problems with positive curvature forms, focusing on Ricci flow.
problem Problems with Riemannian manifolds and positive curvature forms.
method Explains how recent Ricci flow theory can solve these problems.
result Ricci flow is well-suited for solving problems involving positive curvature.
Paper proves properties of minimal hypersurfaces in specific solitons.
problem Characterizing minimal hypersurfaces in shrinking gradient Ricci solitons.
method Analyzes stable minimal hypersurfaces with specific curvature conditions.
result Minimal hypersurfaces in these solitons have zero second fundamental form and normal Ricci curvature.
In this note, we provide some general discussion on the two main versions in the study of Kahler-Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kahler-Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.
Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.
problem Characterize vector fields on hyperbolic spaces Hn that transform them into Ricci-Bourguignon solitons. method Detailed geometric study of vector fields in dimensions n=2,3 and n≥3, focusing on dual forms in odd dimensions. result Dual forms of these vectors are contact forms in odd dimensions.
We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow the Soliton-Ricci flow. It corresponds to a forward Ricci type flow u…
Study stability and instability of Ricci-flat metrics under generalized Ricci flow.
problem Stability and instability of Ricci-flat metrics under Ricci flow.
method Analysis of generalized Ricci flow for Ricci-flat metrics and vanishing 3-forms.
result Dynamical stability and instability results for Ricci-flat metrics and vanishing 3-forms.
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.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
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. Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…
We show that the generalized Kähler-Ricci soliton equation on 4-dimensional toric Kähler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and convex labeled quadrilaterals. In particular, we give the explicit expression of the K…
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequal…
The paper calculates the Chern-Ricci form for a twisted almost Kähler structure.
problem Calculating the Chern-Ricci form for a specific type of almost Kähler manifold.
method Using a twisted almost Kähler structure, the paper derives an explicit formula for the local connection 1-form and calculates the Chern-Ricci form.
result An explicit formula for the local connection 1-form and the Chern-Ricci form of a twisted almost Kähler structure are provided.
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
The paper studies geometric structures in perfect fluid spacetimes with specific metrics.
problem Analyzing the geometric properties of perfect fluid spacetimes with specific metrics.
method Investigates conditions for conformal Ricci-Yamabe soliton and derives Laplace equations.
result Conditions for expanding, steady, or shrinking conformal Ricci-Yamabe solitons are identified.
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
Defines projective Ricci curvature and proves rigidity for sprays.
problem Defining and studying projective Ricci curvature.
method Introduced projective Ricci-flat sprays and studied Randers metrics.
result Global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature.
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.