Holomorphic splitting theorem for Calabi-Yau manifolds with specific properties.
problem Constructing a complete Calabi-Yau metric on a manifold with a specific divisor.
method Solved Monge-Ampère equation on generalized ALG manifolds, used solution to prove holomorphic splitting theorem.
result Proved biholomorphic equivalence of a Calabi-Yau manifold to a product space.
A complex contact structure γ is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Kerγ of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k-vectors on a compl…
We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of C isometrically from the simply-connected Kähler manifold with nonnegative bisectional …
The abstract proves a global splitting theorem for Poisson manifolds.
problem Decomposing compact Kähler Poisson manifolds into simpler components.
method Proving a global splitting theorem using symplectic leaves and finite étale covers.
result Compact Kähler Poisson manifolds can be split into simpler components.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
Cao's splitting theorem says that for any complete Kähler-Ricci flow (M,g(t)) with t∈[0,T), M simply connected and nonnegative bounded holomorphic bisectional curvature, (M,g(t)) is holomorphically isometric to $\C^k\times (N,h(t))$ where (N,h(t)) is a Kahler-Ricci flow with positive Ricci curvature for $t…
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
Let M be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and M admits a nonconstant holomorphic function with polynomial growth, we prove M must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the p…
This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proo…
Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
problem Characterizing holomorphic Lie algebroid connections on Riemann surfaces.
method Analyzes conditions for holomorphic vector bundles to admit Lie algebroid connections based on Lie algebroid properties.
result Conditions for holomorphic vector bundles to admit holomorphic Lie algebroid connections are determined.
In this note, we continue the investigation of a projective Kähler manifold M of semi-negative holomorphic sectional curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of H in the tangent spaces. We prove th…
An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler…
Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.
problem Understanding the geometry of holomorphic Lagrangian fibrations.
method Using special Kähler geometry and parallel splitting of tangent bundles.
result Holomorphic Lagrangian fibrations over non-projective spaces are impossible.
In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…
Study cone structures on contact manifolds to understand their geometric properties.
problem Characterize cone structures on holomorphic contact manifolds.
method Characterize subadjoint varieties among Legendrian submanifolds in terms of contact prolongations.
result Holomorphic horizontal splitting of the canonical distribution on contact G-structures.
New proof of Lorentzian splitting theorems using elliptic operators.
problem Proving splitting theorems in Lorentzian geometry.
method Using a negative homogeneity elliptic p-d'Alembert operator to prove theorems.
result Lorentzian splitting theorems are proven in a framework similar to Riemannian geometry.
Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.
problem Characterize minimal timelike surfaces in R13. method Use a Weierstrass-type formula with holomorphic functions in split-complex numbers to find canonical parameters and corresponding holomorphic functions.
result Enneper surfaces are the only minimal timelike surfaces with polynomial parametrization of degree 3 in isothermal parameters.
Introduces new holomorphic contact structures and proves unobstructedness theorems.
problem Generalizing classical holomorphic contact and symplectic structures.
method Introducing new classes of holomorphic p-contact and s-symplectic manifolds, observing their properties, and proving structure and unobstructedness theorems. result Generalizes classical results on small deformations of complex structures.
New splitting theorem for weighted Finsler spacetimes without Berwald condition.
problem Proving timelike splitting theorems for Finsler spacetimes under weaker conditions.
method Using the p-d'Alembertian and a recently developed strategy. result Established a diffeomorphic splitting for timelike geodesically complete Finsler spacetimes.
We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike spli…
In the current article our primary objects of study are compact complex submanifolds of quotient manifolds of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms. We are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifo…
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
Study Loday algebroids, prove splitting theorem, and linearize problems.
problem Splitting and linearization of Loday algebroids.
method Local splitting-type results, Euler-like derivations.
result Established a general linearization principle.
Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
It is well-known that non-constant holomorphic functions do not exist on a compact complex manifold. This statement is false for a supermanifold with a compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a compact homogeneous complex supermanif…
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for C1 metrics and weights. method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to C1 settings. New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.
problem Understanding the geometry of weighted Lorentz-Finsler manifolds.
method Developed a splitting theorem using weighted Berwald spacetimes and Busemann functions.
result Weighted Lorentz-Finsler manifolds with certain properties split into simpler isometric translations.
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.
Sharp spectral theorem splits certain non-compact manifolds.
problem Proving spectral splitting for non-compact manifolds with specific curvature conditions.
method Sharp spectral analysis and geometric splitting theorem.
result Non-compact manifolds split as RimesN under given curvature constraints. We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positi…
We prove a Lorentzian splitting theorem with weakened curvature conditions.
problem Proving a Lorentzian splitting theorem under weakened Ricci curvature conditions.
method Using achronal limits and geometric maximum principles.
result Strengthened a related result in [29] by removing a boundedness condition on Ricci curvature.
Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.
problem Investigating holomorphic Koszul-Brylinski homology on Poisson manifolds.
method Using Dolbeault cohomology to derive properties of holomorphic Koszul-Brylinski homology.
result Obtained the Leray-Hirsch theorem, Mayer-Vietoris sequence, and Künneth theorem for holomorphic Koszul-Brylinski homology.
Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.
problem Constructing smooth C∗-actions on moduli spaces of super stable curves and maps of genus zero. method Using the implicit function theorem, proving smooth split atlases, and studying automorphism groups.
result Explicit descriptions of normal bundles to fixed loci in terms of spinor bundles and sections.
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.
We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an a…
Study shows mass distribution of random holomorphic sections follows a central limit theorem.
problem Understanding mass distribution of random holomorphic sections.
method Proved a central limit theorem for mass distribution of random holomorphic sections associated with positive line bundles.
result Almost every sequence of random holomorphic sections exhibits quantum ergodicity.
A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Study vortex flows on Riemann surfaces, proving dominated splitting and Anosov properties.
problem Investigate flow properties on Riemann surfaces.
method Associate flow to vortex equations, investigate properties of flow.
result Show that flow always admits a dominated splitting and identify special cases of Anosov flow.
The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alt…
Short proof of Strong Haken Theorem for 3-manifolds.
problem Proving Scharlemann's Strong Haken Theorem for 3-manifolds.
method Short proof using sphere complexes.
result Short proof of Scharlemann's Strong Haken Theorem.
The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.
problem Understanding statistical properties of zeros of random holomorphic sections.
method Proves a central limit theorem for smooth linear statistics of zero divisors of Gaussian sections in line bundles over Kähler manifolds.
result Derives first-order asymptotics and upper decay estimates for Bergman kernels.