The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.
problem Understanding the relationship between Chern and Riemann sectional curvatures on Hermitian manifolds.
method Derivation of Chern sectional curvature expressions and subsequent results on Ricci and scalar curvatures.
result A Hermitian metric is Kähler if and only if its Riemann sectional curvature equals its Chern sectional curvature.
Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.
problem Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds
method Prove the conjecture using curvature identities and properties of Kähler metrics
result Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…
New curvature obstruction for Killing vector fields on Lorentzian manifolds.
problem Existence of timelike or causal Killing vector fields on Lorentzian manifolds.
method New curvature obstruction in terms of timelike or null sectional curvature.
result Extension of Gauss-Bonnet-Chern obstruction to non-zero timelike sectional curvature.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.
problem Proving properties of Sasakian manifolds with negative transverse holomorphic sectional curvature.
method Analyzing the curvature properties and applying the Wu-Yau theorem.
result Compact Sasakian manifolds with negative transverse holomorphic sectional curvature have negative transverse Ricci curvature.
Introduces positivity for (1,1) classes in foliated manifolds.
problem Understanding positivity in foliated manifolds.
method Introduces positivity for a real basic (1,1) class in basic Bott-Chern cohomology group and studies its relationship with negativity of transverse holomorphic sectional curvature. result Establishes the relationship between positivity and negativity in foliated manifolds.
Paper proves Chern flat for 3D Hermitian manifolds with zero real bisectional curvature.
problem Understanding constant curvature Hermitian manifolds in higher dimensions.
method Examined Hermitian threefolds with zero real bisectional curvature, proving Chern flatness.
result Compact Hermitian threefolds with zero real bisectional curvature are Chern flat.
Vanishing theorem for certain tensor fields on compact Hermitian manifolds.
problem Vanishing theorem for holomorphic tensor fields on compact Hermitian manifolds.
method Inspired by X. Yang and L. Ni-F. Zheng's ideas, the proof uses the definiteness of holomorphic sectional curvature.
result Spaces of certain holomorphic tensor fields are trivial under the definiteness of holomorphic sectional curvature.
The paper constructs metrics with negative curvature on complex manifolds.
problem Constructing complete Kähler metrics with negative bisectional curvature on hyperbolic complex manifolds.
method Introducing a mechanism for constructing complete Kähler metrics with negative bisectional curvature.
result Realized Chern slopes c12/c2 for surfaces with negative holomorphic sectional curvature. Researchers confirm conjecture for complex nilmanifolds in higher dimensions.
problem Confirming the conjecture for compact Hermitian manifolds with constant holomorphic sectional curvature.
method Focused on complex nilmanifolds, proving the conjecture for these specific manifolds.
result The conjecture is confirmed for complex nilmanifolds in higher dimensions.
The paper explores mixed curvature for Hermitian manifolds and its implications.
problem Investigating the properties of mixed curvature for Hermitian manifolds.
method Analyzing convex combinations of first Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant mixed curvature are Kähler unless specific conditions are met.
We prove that compact Kähler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit Kähler…
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity KN≤b<0
Develops a graphical calculus for stable curvature invariants.
problem Calculating stable curvature invariants of Riemannian manifolds.
method Graphical calculus based on trivalent graphs with colored edges.
result Derives a curvature identity for compact Einstein manifolds.
We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observa…
The twistor space \Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h_t compatible with the almost complex structures J_1 and J_2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost …
The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.
problem Improving inequalities for Kähler-Einstein manifolds.
method Using invariant theory and curvature conditions to express and improve inequalities.
result Improved inequalities for Kähler-Einstein manifolds with smaller pinching constants.
The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the iso…
Characterizes Kähler-Berwald metrics on complex manifolds.
problem Identifying Kähler-Berwald metrics among strongly convex complex Finsler metrics.
method Geometric characterization using Cartan and Chern-Finsler connections.
result Characterizes Kähler-Berwald metrics in terms of parallelism of the canonical complex structure.
If a normalized Kähler-Ricci flow g(t),t∈[0,∞), on a compact Kähler n-manifold, n≥3, of positive first Chern class satisfies g(t)∈2πc1(M) and has Ln curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will conv…
In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry…
Bounding characteristic numbers of Riemannian manifolds via volume.
problem Bounding characteristic numbers of Riemannian manifolds.
method Using Chern-Weil theory and connections constructed from harmonic metric tensors with bounded Hölder norms.
result Characteristic numbers are bounded proportionally to the volume of Riemannian manifolds.
In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number c1(M)2 is biholomorphic to ${\C}^2$.
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
New curvature K(x) measures manifold properties without integrals.
problem Understanding curvature on compact Riemannian manifolds.
method Developed index expectation curvature K(x) for 2D manifolds, constructed as a product of sectional index expectation curvatures.
result For small 2D manifolds with boundary, definite sign index expectation curvature K(x) exists and satisfies Gauss-Bonnet relation.
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. Study Chern number inequalities for negative curvature Kähler manifolds.
problem Chern number inequalities for compact Kähler manifolds with negative sectional curvature.
method Study L2 ∂ˉildeE-harmonic forms on lifting bundle over universal covering space, observe relationship between Laplace-Beltrami eigenvalues and Euler characteristic. result Euler characteristic inequality involving sectional curvature and Laplace-Beltrami eigenvalues.
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
The paper explores properties of Gauduchon curvature in Hermitian manifolds.
problem Investigating properties of Gauduchon curvature in Hermitian manifolds.
method Analyzing the Ricci curvature of Gauduchon connections and proving existence of metrics.
result Monotonicity theorem for Gauduchon holomorphic sectional curvature.
The article confirms a complex geometry conjecture for a specific type of manifold.
problem Compact Hermitian manifolds with constant holomorphic sectional curvature.
method Restricting to pluriclosed manifolds and confirming the conjecture for Strominger Kähler-like manifolds.
result The conjecture is confirmed for a specific type of Hermitian manifold.
Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.
problem Understanding the distribution of zeros and degeneracy sets of random holomorphic sections.
method Analyzing the pullback of Chern classes and computing currents of integration.
result The limit distribution of zeros of random sections is determined by the Chern form.
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
The article confirms a conjecture for solvmanifolds with complex commutator.
problem Confirming a conjecture about compact Hermitian manifolds with constant holomorphic sectional curvature.
method Analyzing solvmanifolds with complex commutator, extending results on nilmanifolds.
result The conjecture is confirmed for all solvmanifolds with complex commutator.
Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.
problem Calculating curvatures in holomorphic fibrations with degenerate Hermitian forms.
method Theory of Chern connections and curvature forms for degenerate Hermitian forms on holomorphic vector bundles.
result Positive holomorphic sectional curvature in Grassmannian bundles if the base does.
We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bo…
The paper studies constant kth-mixed curvature on Hermitian manifolds and finds self-duality and Kähler conditions.
problem Investigating constant kth-mixed curvature on Hermitian manifolds. method Analyzing Hermitian manifolds with convex combinations of Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant kth-mixed curvature are self-dual, and if k=2, the metric is Kähler. The paper proves a new inequality for CR-warped product submanifolds in complex space forms.
problem Proving a new inequality for CR-warped product submanifolds in complex space forms.
method Developed a first Chen inequality for CR-warped product submanifolds in complex space forms.
result The bound is sharp and uniform in the sign of the holomorphic sectional curvature.
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
The paper proves a new inequality for submanifolds in Riemannian space forms and applies it to find minimal submanifolds.
problem Finding necessary conditions for minimal submanifolds in Riemannian space forms.
method Proving a first Chen inequality for general warped product submanifolds and applying it to derive conditions for minimality.
result A necessary condition for submanifolds to be minimal in Riemannian space forms.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.
problem Finding constant mean curvature surfaces in specific spacetimes.
method Partial differential equations in the complex plane, generalizing holomorphy.
result Extends Hopf's theorem to new spacetime geometries.
Study classifies 4D shrinkers with nonnegative Ricci curvature.
problem Classifying 4D shrinkers with nonnegative Ricci curvature.
method Asymptotic analysis, eigenvalue evolution, Gauss-Bonnet-Chern formula, integration by parts.
result Classifies 4D shrinkers under specific curvature conditions.
Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.
problem Characterizing Hermitian metrics with vanishing second Chern Ricci curvature.
method Analyzing the rigidity of the second Chern Ricci curvature on compact complex manifolds.
result Characterization of second Chern Ricci-flat Hermitian metrics and non-existence results.
Study almost complex structures on six-manifolds using twistor spaces.
problem Understanding the space of almost complex structures on six-dimensional manifolds.
method Using twistor spaces and rational homotopy theory, compute the space of almost complex structures and their homological properties.
result Computed the rational homotopy theoretic minimal model of components of almost complex structures satisfying a Chern number condition.
The paper confirms a conjecture for Bismut torsion parallel metrics.
problem The existence of metrics with constant holomorphic sectional curvature on non-Kähler manifolds.
method Investigation of Bismut torsion parallel metrics.
result The conjecture is confirmed for all non-balanced BTP manifolds.
The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are…
Study Bergman and spectral kernels for non-compact complex manifolds.
problem Analyze asymptotic behavior of kernels over non-compact complex manifolds.
method Generalize scaling method to study Bergman and spectral kernels.
result Derive leading term of Bergman and spectral kernels under local convergence of Chern curvatures.