Statistical manifolds with constant curvature are projectively flat and symmetric.
problem Characterizing statistical manifolds with constant curvature.
method Analyzing the curvature and projective flatness properties of statistical manifolds.
result Statistical manifolds with constant curvature are projectively flat and symmetric.
Study constant Q-curvature metrics on conic 4-manifolds.
problem Find metrics with constant Q-curvature on conic 4-manifolds.
method Analyze related differential equations in the given conformal class.
result Solve constant Q-curvature problem on conic 4-manifolds.
The study classifies quasi-Einstein manifolds with constant scalar curvature.
problem Characterizing quasi-Einstein manifolds with specific curvature properties.
method Classification and construction of examples of quasi-Einstein manifolds.
result Complete classification of quasi-Einstein manifolds with constant scalar curvature.
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
Study existence of conformal metrics with specific curvature properties on compact manifolds.
problem Existence of conformal metrics with constant scalar curvature and boundary mean curvature.
method Proving existence through specific cases and sequences of metrics.
result Existence of conformal metrics in various cases, including positive Yamabe constant.
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
The paper proves compactness for specific surfaces in curved spaces.
problem Compactness of constant mean curvature surfaces in three-manifolds with positive Ricci curvature.
method Proves a compactness theorem with area and genus bounds.
result Lower bound of first eigenvalue for constant mean curvature surfaces.
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
problem Characterizing compact Einstein manifolds with certain curvature operators.
method Bochner technique and Li's result [10]
result Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…
Study on Hermitian metrics and curvature properties of complex manifolds.
problem Analyzing curvature properties of Hermitian metrics on complex manifolds.
method Derivation of formulae and proofs for Chern-Ricci curvatures and holomorphic sectional curvatures.
result Examples of metrics with specific curvature properties.
The paper proves constant mean curvature surfaces in specific manifold types.
problem Existence of surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
method Combines min-max theory with inverse mean curvature flow.
result Existence of compact surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
The paper studies para-Kenmotsu manifolds and their properties.
problem Characterizing and studying properties of para-Kenmotsu manifolds.
method Using tensor equations and curvature conditions to characterize and study properties of para-Kenmotsu manifolds.
result Para-Kenmotsu manifolds with certain curvature conditions are of constant negative curvature −1. Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Th…
Spheres with same curvature in homogeneous 3-manifolds are identical up to isometry.
problem Classifying spheres with constant mean curvature in homogeneous 3-manifolds.
method Proving spheres differing only by isometry and determining curvature values.
result Complete classification of constant mean curvature spheres in homogeneous 3-manifolds.
Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.
problem Creating Kähler metrics with constant scalar curvature on complex manifolds.
method Blowing up the manifold at points and constructing metrics with Poincaré-type singularities.
result Existence of constant scalar curvature Kähler metrics with Poincaré-type singularities.
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Paper builds singular metrics with constant Q-curvature.
problem Constructing metrics with constant Q-curvature on manifolds.
method Utilizes tools from recent years to build weak solutions.
result First construction of singular metrics with positive Q-curvature.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
The study examines properties of f-contact manifolds with constant curvature.
problem Investigating constant curvature in f-contact manifolds. method Analyzing (κ,μ)-nullity condition and f-sectional curvature. result An f-(κ,μ) manifold with constant f-sectional curvature implies specific conditions on μ and κ. This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.
In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.
Paper shows no non-constant harmonic maps under certain curvature conditions.
problem Existence of non-constant harmonic maps between specific manifolds.
method Analyzes curvature conditions and applies rigidity theorems.
result No non-constant harmonic maps exist under specified conditions.
New space for polarized manifolds with constant curvature metrics.
problem Constructing moduli spaces for polarized manifolds.
method Constructing a moduli space with constant scalar curvature Kähler metrics.
result The moduli space admits a natural Kähler metric.
The paper explores constant holomorphic d-scalar curvature on specific manifolds.
problem Existence and prescription of constant holomorphic d-scalar curvature.
method Study of closed, connected almost Hermitian manifolds of dimension n≥6. result Obtained an application and variation formula for a conformal invariant.
Rigidity theorem for manifolds with specific curvature properties.
problem Characterizing complete Riemannian manifolds with vanishing Bach tensor and positive scalar curvature.
method Pointwise inequalities and Sobolev constant inequalities.
result Rigidity results under specific curvature norms.
We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…
Proves inequality for submanifolds with constant mean curvature.
problem Logarithmic Sobolev inequality for submanifolds with constant mean curvature.
method Analyzes submanifolds in manifolds with nonnegative sectional curvature.
result Establishes inequality for submanifolds with constant mean curvature.
The paper proves conditions for stable constant mean curvature surfaces in specific manifolds.
problem Conditions for stable constant mean curvature surfaces in warped product manifolds.
method Analyzes de Sitter-Schwarzschild and Reissner-Nordstrom manifolds, then generalizes to a broader class of three-dimensional warped product manifolds.
result Stable, compact surfaces in specific manifolds are embedded topological spheres.
Classifies surfaces in specific 3-manifolds with isometry group.
problem Classifying surfaces in homogeneous 3-manifolds.
method Explicit classification of four families of surfaces.
result Classification of surfaces in homogeneous 3-manifolds.
This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already car…
Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.
problem Characterizing totally umbilical hypersurfaces in Spin^c manifolds with specific spinor fields.
method Proving constant mean curvature for hypersurfaces carrying parallel, real or imaginary Killing spinors.
result Results extend to Spin^c case, generalizing O. Kowalski's theorem.
Unique conformal metrics found on certain manifolds.
problem Finding unique conformal metrics with constant Q-curvature.
method Proving uniqueness on manifolds with positive scalar curvature.
result Only metrics of the form λg with λ>0 are constant Q-curvature.
The study finds either many or few constant mean curvature hypersurfaces on a manifold.
problem Finding constant mean curvature hypersurfaces on a manifold.
method Analyzing a manifold with a generic Riemannian metric to determine the existence of hypersurfaces.
result Either infinitely many or infinitely many hypersurfaces with specific mean curvatures exist.
In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation…
A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric curvature operator are pointwise constant. It was previously shown that for all n\ge 4, except n=7, any IP manifold either has constant curvature, or is a warped product, with some specific function, of a line and a space of constant curvature…
Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified. The paper studies constant mean curvature hypersurfaces in Finsler manifolds.
problem Understanding geometric properties of hypersurfaces in Finsler manifolds.
method Using volume preserving variation and homothetic navigation.
result Deduced a Heintze-Karcher type inequality and proved an Alexandrov type theorem.
Solves Han-Li conjecture for most compact manifolds.
problem Finding conformal metrics with specific curvature and mean curvature conditions.
method Combining results from Z. C. Han and Y. Y. Li, applying Yamabe and boundary mean curvature conditions.
result Affirmative solution for most cases, except specific conditions.
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
For a compact Riemannian manifold (M,g2) with constant Q-curvature of dimension n≥6 satisfying nondegeneracy condition, we show that one can construct many examples of constant Q-curvature manifolds by gluing construction. We provide a general procedure of gluing together (M,g2) with any compact manifo…
The study proves the existence of certain surface types in 3D spaces.
problem Existence of constant mean curvature surfaces in specific 3D spaces.
method Analysis of homology classes of closed 3-manifolds.
result Existence of constant mean curvature surfaces in specified 3D spaces.
Upper diameter bound for manifolds with positive scalar curvature.
problem Estimating the maximum size of manifolds with positive scalar curvature.
method Proving an upper diameter bound using scalar curvature integral, Yamabe constant, and manifold dimension.
result The power of scalar curvature integral in diameter estimates is sharp and occurs at round spheres with canonical metric.
Existence proven for special metrics on certain complex manifolds.
problem Existence of constant scalar curvature Kähler metrics.
method Compact Kähler manifolds with semi-ample canonical bundles.
result Existence of constant scalar curvature Kähler metrics proven.