Study shows almost complex structures with certain tensor properties are prevalent.
problem Characterizing almost complex structures with specific tensor properties.
method Analyzes the space of almost complex structures on compact manifolds.
result The space of almost complex structures with rank at least k Nijenhuis tensor is either empty or dense in each component.
The paper proves conditions for almost complex manifolds to admit almost complex structures through connected sums.
problem Conditions for almost complex manifolds to admit almost complex structures via connected sums.
method Proving conditions for almost complex manifolds to admit almost complex structures through connected sums of specific manifolds.
result Conditions for the existence of almost complex structures on connected sums of specific manifolds.
Study on invariant almost complex structures on real flag manifolds.
problem Existence of invariant almost complex structures on real flag manifolds.
method Analysis of real flag manifolds associated to split real forms of complex simple Lie algebras.
result Some real flag manifolds do not admit invariant almost complex structures.
Study on biharmonic almost complex structures on compact manifolds.
problem Existence and regularity of biharmonic almost complex structures.
method Analyzes biharmonic almost complex structures on compact almost Hermitian manifolds, focusing on dimension four.
result Existence of energy-minimizing biharmonic almost complex structures for various topologies and homotopy classes.
Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.
problem Understanding the Kodaira dimension of almost complex manifolds with SU(m)-structures.
method Introduced almost complex structure of splitting type and associated SU(m)-structure. Provided constructions for non-invariant almost complex structures with specific Kodaira dimensions.
result Found non-invariant almost complex structures with Kodaira dimensions 0 and -∞.
Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
Solves Monge-Ampère on non-integrable manifolds.
problem Existence and uniqueness of solutions to Monge-Ampère equation on non-integrable almost complex manifolds.
method Existence and uniqueness proved using the Monge-Ampère equation.
result Existence and uniqueness of solutions proved.
Almost complex structures found on many homotopy complex projective spaces.
problem Finding almost complex structures on homotopy complex projective spaces.
method New proof using Chern classes and homotopy properties.
result Classification of almost complex structures on homotopy CPn for 3≤n≤6. The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant B-transformations and classification of structures. result No GM2-maximal real flag manifolds admit integrable invariant generalized almost complex structures. Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.
problem The study of canonical almost complex structures on ACH Einstein manifolds.
method Variational problem with the Dolbeault Laplacian acting on (0,1)-forms. result Deformation results of Einstein ACH metrics associated with critical almost complex structures.
Study functionals on almost complex structures for Yau's Challenge.
problem Yau's Challenge on compact C-manifolds. method Variational properties of functionals on almost complex structures.
result Variational properties could be used to tackle Yau's Challenge.
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
The paper proves smoothness of weakly biharmonic almost complex structures in dimension four.
problem Existence and regularity of weakly polyharmonic almost complex structures.
method Elliptic system analysis and critical growth nonlinearities.
result Weakly biharmonic almost complex structures are smooth in dimension four.
In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group G2. While he did not solve the (currently still open) problem of determining whether there exists an int…
We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by dis…
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
No almost complex structures on the six-sphere satisfy certain commutation conditions.
problem Existence of almost complex structures on the six-sphere with specific properties.
method Refined differential inequalities and almost-complex analogue of the Gauss map.
result No integrable almost complex structures on the six-sphere satisfy the given conditions.
Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of (R4,J), for some almost complex structure J if and only if it is an elliptic curve. Furthermore we show that any (almost) complex 2n-torus can be holomorphically embedded in (R4n,J) for a suitable almo…
The study shows conditions for almost complex structures on rational homology spheres and limits on their Betti numbers.
problem Conditions for almost complex structures on rational homology spheres.
method Analyzes rational homology spheres and provides examples of manifolds with almost complex structures.
result Closed almost complex manifolds with sum of Betti numbers three have dimensions that are powers of two.
An \emph{ω-admissible almost complex structure} on a 2n-dimensional symplectic manifold (M,ω) is a ω-calibrated almost complex structure J admitting a nowhere vanishing ∂ˉJ-closed (n,0)-form ψ. After giving some examples we consider the moduli space of admissible almost complex structures a…
Survey on harmonic almost complex structures on Riemannian manifolds.
problem When an almost complex structure is harmonic.
method Analysis of old and new results on compatibility and mappings.
result Special attention to Atiyah-Hitchin-Singer and Eells-Salamon structures.
We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.
Study on Kodaira dimension of specific solvmanifolds without complex structures.
problem Analyzing Kodaira dimension for almost complex 4D solvmanifolds without integrable structures.
method Classification of solvmanifolds and computation of Kodaira dimension for specific structures.
result Showed that Kodaira dimension is not a deformation invariant for some solvmanifolds.
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
New method shows almost complex structures can be approximated smoothly.
problem Approximating almost complex structures on manifolds.
method Combining h-principle techniques with directed immersion relation.
result Smooth approximation of almost complex structures with controlled error.
We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.
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.
In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongatio…
Paper reviews spheres with almost complex structures.
problem Identifying spheres with almost complex structures.
method Characteristic classes and Bott periodicity theorem.
result Only S^2 and S^6 admit almost complex structures.
The paper studies a heat flow for almost complex structures and proves convergence under certain conditions.
problem The study of harmonic heat flow for almost complex structures compatible with a Riemannian metric.
method Definition and analysis of the harmonic heat flow, proving existence and convergence under small energy conditions.
result The flow converges to a Kähler structure if the initial energy is small, but there are finite time singularities for small enough initial energy.
Study of special almost complex structures on symplectic manifolds.
problem Understanding special almost complex structures on symplectic manifolds.
method Examined homogeneous compatible almost complex structures, focusing on those with Chern-Ricci form being a multiple of the symplectic form.
result Compact isotropy co-adjoint orbits of semi-simple Lie groups admit special compatible almost complex structures under certain conditions.
New pseudo-Kähler Einstein spaces found with special almost complex structures.
problem Identifying new pseudo-Kähler Einstein spaces with special almost complex structures.
method Examining 4-symmetric symplectic spaces with compatible almost complex structures.
result Found pseudo-Kähler Einstein spaces with an integrable and non-integrable pair of compatible almost complex structures.
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
Study anti-invariant cohomology on almost complex manifolds, showing infinite and finite dimensions.
problem Understanding the cohomology of anti-invariant forms on almost complex manifolds.
method Construction of specific almost complex structures and analysis of cohomology groups.
result Found manifolds with anti-invariant cohomology of infinite, 0-1, and 2 dimensions.
Paper calculates curvatures of metrics from isotropic structures.
problem Calculating curvatures of metrics induced by isotropic structures.
method Calculating curvature tensors of induced Riemannian metrics.
result Curvatures of the metrics are determined.
Generically an almost complex structure has no symmetries at all, but there exist symmetric structures. In this paper we describe how to guarantee that the pseudogroup of local symmetries is small (finite-dimensional). It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some …
The space A of almost complex structures on a closed manifold M is studied. A natural parametrization of the space A is defined. It is shown, that A is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space A is found. The space ${\…
We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins with a discussion of basic geometry of almost complex 6-manifolds. …
Study of complex surfaces in a specific pseudo-Riemannian space.
problem Classifying complex surfaces in a particular pseudo-Riemannian space.
method Investigation of totally geodesic and parallel second fundamental form complex surfaces.
result Complete classification of totally geodesic and parallel second fundamental form complex surfaces.
We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existen…
The paper explores 6D almost complex structures with maximal symmetry groups.
problem Finding the largest automorphism groups of 6D almost complex structures.
method Analyzing the Nijenhuis tensor and automorphism groups of 6D almost complex structures.
result The largest automorphism group dimension is 10, realized by 3 strictly nearly Kähler spaces.
Computational techniques calculate dimensions of complex structures.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
We prove that any quasitoric manifold M2n admits a Tn-invariant almost complex structure if and only if M admits a positive omniorientation. In particular, we show that all obstructions to existence of Tn-invariant almost complex structure on M2n arise from cohomology of underlying polytope - and henc…