Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
Classifies Kähler manifolds with special gradient vector fields.
problem Characterizing Kähler manifolds with geodesic holomorphic gradients.
method Classification based on specific vector fields and integrability conditions.
result All such manifolds are biholomorphic to bundles of complex projective spaces.
The paper proves compactness for holomorphic curves with boundary on nearby Lagrangians.
problem Compactness of holomorphic curves with boundary on nearby Lagrangians.
method Generalizes earlier work on compactness, proving a limit configuration of holomorphic curves joined by gradient flow lines.
result Exponential estimate analyzing the interface between holomorphic parts and gradient flow lines.
The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
problem Conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
method Investigation of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
result Identification and determination of weight functions with real holomorphic gradient fields on specific metrics.
The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.
problem Estimating dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
method Proved upper bounds for dimensions and existence of sections using polynomial growth.
result Upper bounds for dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
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.
The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic las…
Holomorphic functions on certain manifolds are isometric to Lie groups.
problem Characterizing holomorphic functions on specific Hermitian manifolds.
method Gradient estimate for holomorphic functions, sub-Riemannian geometry.
result Universal cover of complete Hermitian manifolds with flat Chern connection is holomorphically isometric to a complex Lie group.
Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…
Study heat flow for gauged holomorphic maps over Kähler manifolds.
problem Understanding the moduli space of gauged holomorphic maps.
method Gradient flow approach inspired by Atiyah and Bott, heat flow method.
result Global existence of smooth solutions of gradient flow equations.
The paper studies convergence of a flow related to Yang-Mills-Higgs equations on holomorphic pairs.
problem Analyzing convergence of a flow related to Yang-Mills-Higgs equations on holomorphic pairs.
method Study of the negative gradient flow of the Yang-Mills-Higgs functional on holomorphic pairs (A,u). result Uniform convergence of the negative gradient flow in the W1,2imesW2,2-topology. Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.
problem Understanding polynomial growth of holomorphic functions on Kähler-Ricci shrinkers.
method Analyzing scalar curvature conditions to prove finite generation of the ring of holomorphic functions.
result The ring of holomorphic functions with polynomial growth on Kähler-Ricci shrinkers is finitely generated.
The fundamental properties of J-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a J-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete Kähler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of cur…
The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…
Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.
problem Gromov's non-squeezing theorem in symplectic geometry.
method Reparametrization of pseudo-holomorphic curves and application of mean value inequality or Gromov-Schwarz lemma.
result Uniform bounds on the gradient of pseudo-holomorphic curves leading to compactness of moduli space.
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.
By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtaine…
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…
We study complete noncompact long time solutions (M,g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at (p,t) for all t for some c>0, then there always e…
A 2-category categorifies complex Lagrangians in hyperkähler manifolds.
problem Categorify the Fukaya category of hyperkähler manifolds.
method Formalizes morphisms in a 2-category based on Fueter maps.
result Fueter maps correspond to complex gradient trajectories in cotangent bundles.
We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler potential of the soliton will converge to the original soliton under Kaehler-Ricci …
Novel multisymplectic framework for pseudo-Fueter curves in Hamiltonian field theory.
problem Generalizing Floer theory to multisymplectic geometry.
method Introducing pseudo-Fueter curves in a compatible almost hyperkähler structure.
result Gradient lines of multisymplectic action functional are pseudo-Fueter curves.
We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps H(P,X), where P is a principal bundle on a Riemann surface Σ and X is a Kähler Hamiltonian G-manifold. For compact Σ, possibly with boundary, we prove long time existence of the gradient flow. …
Study describes Yang-Mills flow asymptotics on Riemann surfaces.
problem Analyzing Yang-Mills flow on Riemann surfaces.
method Monotonicity property of the flow, applied in arbitrary dimensions.
result Complete description of Yang-Mills flow asymptotics.
This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in…
The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.
problem Understanding the geometry and topology of Sasaki-Ricci solitons.
method Analyzing the properties of complete gradient shrinking Sasaki-Ricci solitons, proving connectedness at infinity and compactness under certain curvature conditions.
result Proves that Sasaki-Ricci solitons are either connected at infinity or compact, generalizing results from previous studies.
In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and…
The paper studies gradients of geodesic-length functions and systoles on Teichmüller spaces.
problem Understanding the behavior of geodesic-length functions and systoles on Teichmüller spaces.
method Analyzing the Lp-norms of gradients of geodesic-length functions along systolic curves. result The Lp-norms of gradients of geodesic-length functions are uniformly comparable to the systole. Suppose that a polarised Kähler manifold (X,L) admits an extremal metric ω. We prove that there exists a sequence of Kähler metrics {ωk}k, converging to ω as k→∞, each of which satisfies the equation ∂ˉgradωk1,0ρk(ωk)=0; the (1,0)-part of the gradient of the B…
The study of constant pluriharmonic functions on Kähler-Ricci solitons.
problem Conditions for pluriharmonic functions to be constant on Kähler-Ricci solitons.
method Introducing a holomorphic quantity and investigating gradient integrability.
result Conditions for pluriharmonic functions to be constant on Kähler-Ricci solitons.
The paper describes orbits of parabolic subgroups in complexified actions.
problem Understanding orbits of parabolic subgroups in complexified actions.
method Analyzes the gradient map μₚ to describe orbits of parabolic subgroups.
result Describes compact orbits of parabolic subgroups in terms of the gradient map.
For a Kähler manifold endowed with a weighted measure e−fdv, the associated weighted Hodge Laplacian Δf maps the space of (p,q)-forms to itself if and only if the (1,0)-part of the gradient vector field ∇f is holomorphic. We use this fact to prove that for such f, a finite energy f harmonic …
Paper solves a singular version of Gauduchon's conjecture.
problem Finding Gauduchon metrics with prescribed Ricci curvature on compact complex manifolds.
method Study of the Monge-Ampère equation for (n−1)-plurisubharmonic functions with a gradient term, adapted to singular settings. result Obtained a C0-estimate for the singular problem, proving smoothness of solutions on holomorphic Kähler families. This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…
Study on semistable points and convexity of gradient maps for group actions.
problem Analyzing semistable points and convexity in group actions.
method Examining a real reductive group action on a Kahler manifold with Hamiltonian properties.
result Openness and connectedness of semistable points, convexity theorems for G-action and two-orbit variety. Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact Kähler manifolds as well as a short time existence of the negative …
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.
Holomorphic Jacobi manifolds integrate to complex contact groupoids.
problem Integrating holomorphic Jacobi manifolds.
method Homogenization scheme to identify and integrate holomorphic Jacobi manifolds to complex contact groupoids.
result Holomorphic Jacobi manifolds integrate to complex contact groupoids.
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.
We introduce holomorphic Riemannian maps between almost Hermitian manifolds as a generalization of holomorphic submanifolds and holomorphic submersions, give examples and obtain a geometric characterization of harmonic holomorphic Riemannian maps from almost Hermitian manifolds to Kaehler manifolds.
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.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.