Study on slant submanifolds in complex space forms with a new Simons' type formula.
problem Characterizing slant submanifolds in complex space forms.
method Developed an integral formula of Simons' type for Kaehlerian slant submanifolds.
result Proved a main result using the new Simons' type formula.
Study curvature inequalities for real hypersurfaces in complex space forms.
problem Understanding curvature properties of real hypersurfaces in complex space forms.
method Established an inequality relating Ricci curvature, mean curvature, and normal curvature; classified hypersurfaces achieving equality.
result Classified real hypersurfaces in two-dimensional non-flat complex space forms achieving equality in curvature inequality.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.
Study ruled real hypersurfaces in nonflat complex space forms with constant norm shape operators.
problem Proving nonexistence and characterizing ruled hypersurfaces in nonflat complex space forms.
method Analyzing shape operators with constant squared norm and using biharmonic properties.
result Biharmonic ruled real hypersurfaces are minimal, leading to their classification.
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.
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
Study on PMC surfaces in complex space forms, linking biconservative and totally real properties.
problem Characterizing PMC surfaces in complex space forms and their properties.
method Analyzing interactions between PMC, totally real, and biconservative properties; proving rigidity and reduction codimension results.
result PMC surfaces in non-flat complex space forms are biconservative if and only if totally real.
Study quasi-Einstein metrics on real hypersurfaces of complex space forms.
problem No Einstein metrics in non-flat complex space forms, explore quasi-Einstein condition.
method Analyze real hypersurfaces in complex Euclidean space, classify quasi-Einstein metrics.
result Classify quasi-Einstein metrics on real hypersurfaces of complex Euclidean space.
Study on real hypersurfaces with special solitons in complex space forms.
problem Characterizing real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. method Analyzing real hypersurfaces with ∗-Ricci solitons where the potential vector field is in the principal curvature space and holomorphic distribution. result Characterized real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. New algebra structure for curvature measures in complex space forms.
problem Understanding curvature measures in complex space forms.
method Explicitly describing the algebra structure of dual unitarily invariant curvature measures.
result Characterization of invariant valuations on complex space forms.
Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the…
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
The paper classifies product minimal Lagrangian submanifolds in complex space forms.
problem Understanding minimal Lagrangian submanifolds in complex space forms.
method Examining submanifolds as Riemannian products with constant sectional curvature.
result Complete classification of product minimal Lagrangian submanifolds.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.
problem Characterizing spheres in complex and Sasakian space forms.
method Establishing optimal integral inequalities involving Ricci curvature and second fundamental form norms.
result New characterizations of Whitney and contact Whitney spheres in complex and Sasakian space forms.
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
Paper classifies special slant surfaces with varying curvature.
problem Classifying surfaces with non-constant mean curvature.
method Analyzing special slant surfaces in complex space forms.
result Complete classification of surfaces with non-constant mean curvature.
The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter…
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
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.
The paper studies extremal Kaehler metrics induced by complex space forms, proving properties in both finite and infinite dimensions.
problem Analyzing extremal Kaehler metrics induced by complex space forms in finite and infinite dimensions.
method Proving properties of extremal Kaehler metrics under specific conditions in both finite and infinite dimensional settings.
result Extremal Kaehler metrics induced by infinite dimensional elliptic complex space forms have constant non-positive holomorphic sectional curvature under stability conditions.
Simple proof for special surface classification.
problem Classifying surfaces with specific curvature properties.
method Elementary proof for parallel mean curvature surfaces.
result A lemma is proven for surfaces in complex space forms.
The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denot…
We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
The paper studies real hypersurfaces in complex space forms with Miao-Tam critical metrics.
problem Characterizing real hypersurfaces with Miao-Tam critical metrics in complex space forms.
method Analyzing the equation −(Δgλ)g+ablag2λ−λRic=g for real hypersurfaces in complex space forms. result Compact real hypersurfaces in complex Euclidean space with Miao-Tam critical metrics are spheres, and non-flat complex space forms do not admit such metrics.
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
The study finds minimal surfaces in complex space forms are often totally geodesic.
problem Characterizing minimal surfaces with specific geometric properties in complex space forms.
method Analyzing free-boundary minimal surfaces in geodesic balls of complex space forms.
result Minimal surfaces in certain complex space forms are either totally geodesic or superminimal.
We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.
Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or H-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-sp…
The paper proves existence and uniqueness of slant immersions in complex space forms.
problem Existence and uniqueness of slant immersions in complex space forms.
method Established existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
result Existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.
Paper establishes new inequality for Riemannian maps and applies it to various space forms.
problem Developing a new inequality for Riemannian maps and its applications.
method Proposed and utilized a general Chen's first inequality for Riemannian maps and applied it to various space forms.
result Validated the new inequality and compared results with existing approaches.
The paper establishes new Casorati inequalities for various Riemannian maps and submersions.
problem Developing new inequalities for Riemannian maps and submersions.
method Using general forms of Casorati inequalities, the paper derives inequalities for specific Riemannian spaces.
result The paper provides new Casorati inequalities for Riemannian maps and submersions.
In this paper the result of real hypersurfaces in non-flat complex space forms, whose structure vector field ξ belongs to the κ-nullity distribution is extended in case of three dimensional real hypersurfaces in non-flat complex space forms. Furthermore, generalization of notion (κ,μ)-nullity distribution defin…
Calabi developed a tool to classify Kähler immersions into complex space forms.
problem Classifying Kähler immersions of Kähler manifolds into complex space forms.
method Calabi introduced a diastasis function to establish necessary and sufficient conditions for local Kähler immersions.
result Calabi's criterion for Kähler immersions is powerful but often difficult to apply.
The paper studies complex space forms via Laplace operators and curvature properties.
problem Characterizing \K manifolds with specific Laplace operator properties.
method Analyzing \K manifolds satisfying the Δ-property and proving curvature implications.
result Complex space forms are characterized by the Δ-property and curvature parallelism.
This paper proves a conjecture about Kähler manifolds and complex space forms.
problem Proving Kähler manifolds satisfying the Δ-property are complex space forms.
method Analyzing Kähler Laplacian and complex Euclidean Laplacian properties.
result Validates the conjecture that Kähler manifolds satisfying the Δ-property are complex space forms.
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.
The paper classifies Lagrangian submanifolds in complex space forms achieving a specific curvature inequality.
problem Classifying Lagrangian submanifolds in complex space forms that achieve a specific curvature inequality.
method Analyzing the equality case of a curvature inequality for Lagrangian submanifolds in complex space forms.
result Classification of δ(2,n−2)-ideal Lagrangian submanifolds in n-dimensional complex space forms. 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…
Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.
In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.
The paper defines pseudo-Einstein real hypersurfaces in complex space forms.
problem Characterizing pseudo-Einstein real hypersurfaces in complex space forms.
method Analyzing the Ricci tensor properties of real hypersurfaces.
result A real hypersurface is pseudo-Einstein if and only if its Ricci tensor satisfies a specific form.
In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two…