New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate L2-norm of holomorphic sectional curvature. result Holomorphic sectional curvature fully determines the curvature tensor.
Classifies Kähler metrics with constant holomorphic curvature.
problem Classifying Kähler metrics with constant holomorphic sectional curvature.
method Exploiting the geometry of the bundle of 1-jets of holomorphic functions.
result Local classification of Kähler metrics with constant holomorphic sectional curvature.
Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
problem Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
method Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations using specific curvature conditions.
result Explicit construction of Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
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.
The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…
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 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…
The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.
problem Proving the existence of complete Kähler metrics with negative holomorphic bisectional curvature in certain domains.
method Analyzing bounded domains in Cn with specific curvature properties. result Strictly pseudoconvex bounded domains and domains with squeezing function tending to 1 at boundary points admit complete Kähler metrics with negative holomorphic bisectional curvature everywhere.
The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.
problem Characterizing compact Kähler manifolds with nonnegative holomorphic sectional curvature.
method Holonomy principle and geometric properties.
result Compact Kähler manifolds with nonnegative holomorphic sectional curvature are projective and rationally connected.
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.
Directly proves Wu's theorem on negative curvature metrics.
problem Subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
method Quick direct proof
result Subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
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.
Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.
problem Proving positive curvature of blowups of manifolds with positive holomorphic sectional curvature.
method Calculates curvature tensor of a specific metric on the blowup's exceptional divisor.
result Holomorphic sectional curvature is negative in some directions for small enough t.
In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant δ depending on the ratio such that $\cosα\ge…
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
The paper proves a Schwarz lemma for mappings between specific types of manifolds.
problem Establishing a Schwarz lemma for mappings between Kähler and complex Finsler manifolds.
method Using properties of holomorphic sectional curvature and radial sectional curvature.
result A Schwarz lemma for holomorphic mappings between Kähler and complex Finsler manifolds.
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension −∞. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…
Estimates curvature for holomorphic maps on Riemann surfaces.
problem Curvature estimation for holomorphic maps on open Riemann surfaces.
method Use of jet differentials to establish a Gauss curvature estimate.
result Established a Gauss curvature estimate for holomorphic maps.
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
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
The study proves the non-existence of certain Kähler metrics with specific curvature properties.
problem Non-existence of complete Kähler metrics with negatively pinched holomorphic sectional curvature.
method Construction of a Kähler metric with negatively pinched holomorphic sectional curvature and application of equivalence of invariant metrics.
result The dichotomy of completeness and non-existence of Kähler metrics with negatively pinched holomorphic sectional curvature.
The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomor…
In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.
The aim of this paper is to describe Kahler surfaces with quasi-constant holomorphic curvature
Study curvature in holomorphic fibration fields.
problem Curvature operator in Bergman spaces.
method Careful study of curvature operator in Kähler manifolds.
result Detailed analysis of curvature in smoothly bounded pseudoconvex domains.
The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.
Sharp lower bound for curvature in Kähler manifolds.
problem Curvature bounds for real hypersurfaces in Kähler manifolds.
method Gauss equation for semi-isometric CR immersions.
result Proves $ rac12$-positivity of Tanaka-Webster scalar curvature.
In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio λ of the maximum and the minimum of the holomorphic sectional curvatures <2, then there exists a positive constant $δ>\frac{29(λ-1)}{\sqrt{(48-24λ)^{2}+(29λ-29…
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
The study characterizes symmetries in Kaehler manifolds.
problem Understanding symmetries in Kaehler manifolds.
method Analyzing specific types of Kaehler manifolds: constant holomorphic sectional curvature, semisymmetric, and holomorphically pseudosymmetric.
result Characterization results and geometric interpretation of the complex Tachibana tensor.
The main result of this note is that, for each n∈{1,2,3,…}, there exists a Hodge metric on the n-th Hirzebruch surface whose positive holomorphic sectional curvature is (1+2n)21-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address th…
Analytic plane curves determine unique conformal coordinates.
problem Determining a conformal coordinate system for analytic plane curves.
method Holomorphic continuation of the Frenet curvature form.
result Holomorphic continuation of the curvature form uniquely determines a conformal coordinate net.
In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…
Schwarz lemma extended to equality cases and curvature on manifolds.
problem Extending Schwarz lemma to equality cases and studying curvature.
method Analyzing Schwarz lemma inequalities and equalities, studying holomorphic sectional curvature.
result Holomorphic maps are totally geodesic and have constant rank when Schwarz lemma equality holds.
The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
In this paper, the theory of functions of one complex variable is explored to study linearly full unramified holomorphic two-spheres with constant curvature in G(2,n) satisfying that the generated harmonic sequence degenerates at position 2. Firstly, we determine the value distribution of the curvature and give the…
We study Kaehlerian manifolds with Norden metric g and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of (R2n+2,g,J) with constant totally real sectional curvatures.
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…
Study of holomorphic curves and surfaces using singularity theory.
problem Understanding the geometry of holomorphic curves and complex surfaces.
method Application of singularity theory to holomorphic curves and surfaces.
result Definition of geometric invariants for curves and surfaces.
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.
On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional…
Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.