Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
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.
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
Holomorphic maps of degree one are biholomorphic, confirming a partial order.
problem Understanding the partial order of holomorphic maps.
method Analyzing holomorphic maps of positive degree between compact complex manifolds.
result Holomorphic maps of degree one are biholomorphic.
Study proper holomorphic maps between symmetric domains, proving rigidity and isometric properties.
problem Characterizing proper holomorphic maps between symmetric domains.
method Analyzing maps between bounded symmetric domains of the same rank, proving rigidity and isometric properties.
result Proper holomorphic maps are totally geodesic isometric embeddings under certain conditions.
Holomorphic maps between moduli spaces are shown to be forgetful for large g.
problem Characterizing holomorphic maps between moduli spaces.
method Proving that only forgetful maps are non-constant for large g.
result Forgetful maps are the only non-constant holomorphic maps between moduli spaces for g≥4. No non-constant holomorphic maps between certain complex manifolds with specific properties.
problem Existence of non-constant holomorphic maps between complex manifolds.
method Analyzing properties of tangent and cotangent bundles, pseudo-effectiveness, and nefness.
result Holomorphic maps between certain complex manifolds are constant.
Formula calculates residues for maps near holomorphic distributions.
problem Calculating residues for maps near holomorphic distributions.
method Residues formula for maps generically transversal to regular holomorphic distributions.
result Established a residues formula for maps near holomorphic distributions.
In this paper, we study deformations of holomorphic Poisson maps which extend Horikawa's series of papers on deformations of holomorphic maps in the context of holomorphic Poisson deformations. In appendices, we present deformations of Poisson morphisms in the language of functors of Artin rings which is the algebraic …
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
Study finite rank bundle over J-Holomorphic map moduli spaces with exponential decay.
problem Understanding the structure of J-Holomorphic map moduli spaces.
method Prove exponential decay of derivative of gluing maps for a finite rank bundle.
result Exponential decay of the derivative of the gluing maps for the finite rank bundle.
H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite energy foliations and the Weinstein conjecture, as well as in folded holomorphic …
Integral inequalities for holomorphic maps prove rigidity and degeneracy theorems.
problem Rigidity and degeneracy theorems for holomorphic maps without curvature sign assumptions.
method Integral inequalities derived from holomorphic maps between complex manifolds.
result Proves rigidity and degeneracy theorems for holomorphic maps.
Two new boundary Schwarz lemmas for holomorphic maps established.
problem Establishing boundary Schwarz lemmas for holomorphic maps.
method Two boundary versions of the Schwarz lemma for holomorphic maps.
result First boundary Schwarz lemmas for general holomorphic self maps and biholomorphisms with invariant Kähler metrics.
Local holomorphic maps preserving (p,p) forms are shown to be isometries.
problem Preserving (p,p) forms under holomorphic maps between Kähler manifolds.
method Analyzing local holomorphic maps between Kähler manifolds, proving isometries up to scalars.
result Holomorphic maps preserving (p,p) forms are isometries under certain conditions.
Paper derives formulas for holomorphic maps between Hermitian manifolds and proves related theorems.
problem Holomorphic maps between Hermitian manifolds and their properties.
method Derives ∂∂-Bochner formulas and proves Schwarz lemma type estimates. result Generalizes Ni's results to Hermitian manifolds, proving rigidity and degeneracy theorems.
The paper establishes Schwarz type lemmas for pseudo-Hermitian manifolds.
problem Understanding the geometry and mappings of pseudo-Hermitian manifolds.
method Using sub-Laplacian or Hessian type Bochner formulas and comparison theorems.
result Established Schwarz type results for \emph{CR} maps and transversally holomorphic maps.
The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.
problem Understanding the factorization of harmonic maps between Riemann surfaces and manifolds.
method The proof relies on geometric properties of the Hopf differential and properties of holomorphic and anti-holomorphic diffeomorphisms.
result The theorem provides a factorization of harmonic maps under certain conditions involving holomorphic or anti-holomorphic diffeomorphisms.
Proves rigidity of maps between balls with Hölder boundary continuity.
problem Rigidity of proper holomorphic maps between unit balls with Hölder boundary continuity.
method Proves rigidity for maps with symmetries and Hölder boundary continuity.
result Proves rigidity for maps with Hölder exponent > 1/2 on the boundary.
The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
The paper classifies holomorphic maps between Riemann surface configuration spaces.
problem Classifying holomorphic maps between configuration spaces of Riemann surfaces.
method Group-theoretic rigidity results promoted to the space level.
result Complete classifications of holomorphic maps between configuration spaces of Riemann surfaces.
The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.
problem Metric inequivalence and characterization of proper holomorphic maps.
method Explicit characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
result Characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
Study Dirac-harmonic maps on Riemann surfaces and their relation to J-holomorphic curves.
problem Understanding critical points of fermionic action functionals on Riemann surfaces.
method Analyzing Dirac-harmonic maps and their relation to J-holomorphic curves on Kaehler manifolds.
result The tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps.
Study the asymptotics of holomorphic maps from polydisk to disk.
problem Understanding the asymptotics of holomorphic maps from polydisk to disk.
method Analyzing the conjugation action of a unipotent subgroup of PSL2(R).
result Results in the asymptotics of the translation flow on holomorphic maps.
Study proper holomorphic maps between specific domains, proving rigidity under certain conditions.
problem Proper holomorphic maps between type-I irreducible bounded symmetric domains. method Analyzing maps under specific assumptions, using automorphisms and a defined map Gh. result Rigidity results for maps, showing conditions on dimensions and existence of automorphisms.
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
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.
The paper proves that certain maps from complex space to Kahler manifolds are holomorphic.
problem Characterizing harmonic maps from complex space to Kahler manifolds.
method Proves that harmonic maps from Cn to any Kahler manifold must be holomorphic under a specific condition. result Harmonic maps from Cn to Kahler manifolds are holomorphic under an energy density assumption. We classify all tight holomorphic maps between Hermitian symmetric spaces of non-compact type.
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.
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
problem Analyzing holomorphic maps between pseudo-Hermitian and Hermitian manifolds.
method Using Bochner formulas and comparison theorems.
result Established Schwarz type lemmas for holomorphic maps.
The paper discusses properties of holomorphic one-forms on certain complex manifolds.
problem Analyzing holomorphic one-forms on weakly 1-complete manifolds.
method Examining connectivity of pairs and criteria for proper holomorphic mappings.
result Criteria for proper holomorphic mappings onto Riemann surfaces.
Study generalizes map properties between Hermitian manifolds preserving specific forms.
problem Understanding maps between Hermitian manifolds that preserve certain forms.
method Generalizing results from Chan-Yuan [2025] to new maps.
result Obtained further rigidity and non-existence theorems.
Study on harmonicity of maps between different types of almost contact metric manifolds.
problem Understanding harmonicity of maps between various almost contact metric manifolds.
method Analyzing and deriving new results for different subclasses of almost contact metric manifolds.
result Obtained new results and recovered, generalized, and corrected known results.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
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.
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuou…
Study defines and proves cobordism maps on periodic Floer homology using Seiberg-Witten and holomorphic curve methods.
problem Understanding cobordism maps on periodic Floer homology.
method Defined cobordism maps via Seiberg-Witten theory and holomorphic curves. Proved equivalence of definitions under certain conditions.
result Equivalence of two definitions of cobordism maps on periodic Floer homology.
Extends holomorphic functions on complex manifolds to larger spaces.
problem Extending holomorphic functions on complex manifolds.
method Proving the existence of a larger space B(S,X) for continuous maps that allows holomorphic continuation. result Bounded holomorphic functions on C(S,X) can be extended to holomorphic functions on B(S,X). Study holomorphic sections of Deligne-Hitchin space, answering Simpson's question.
problem Understanding holomorphic sections of Deligne-Hitchin moduli space.
method Investigate sections invariant under anti-holomorphic involutions and relate to harmonic maps.
result Answered Simpson's question on holomorphic sections.
Constructs the moduli space of super J-holomorphic curves.
problem Defines and constructs the moduli space of super J-holomorphic curves.
method Uses component fields of a super differential equation and a transversality argument.
result Constructs the moduli space of super J-holomorphic curves as a smooth subsupermanifold.
Proves singularities of codimension one objects under finite holomorphic maps.
problem Analyzing singularities of codimension one objects under finite holomorphic maps.
method Generalizes previous results by proving the singularity of pullbacks of singular codimension one holomorphic foliations.
result The preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map is again singular.
The paper explores anti-hyperbolicity for hyperkähler varieties.
problem Anti-hyperbolicity of hyperkähler varieties.
method Exploring various examples and criteria for meromorphic and holomorphic dominability by C^m.
result Generalizing known results about K3 surfaces to hyperkähler manifolds.
In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…
Let M be a n-dimensional complex manifold and f,g:M→M two distinct holomorphic self-maps. Suppose that f and g coincide on a globally irreducible compact hypersurface S⊂M. We show that if one of the two maps is a local biholomorphism around S′=S−Sing(S) and, if needed, S′ sits into M …
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …