Study flows on complex surfaces to find weak hyperKähler structures.
problem Classifying nondegenerate generalized Kähler surfaces.
method Generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure.
result Long time existence and convergence to weak hyperKähler structure.
Study non-degenerate singular points of Poisson-Nijenhuis structures.
problem Non-degenerate singular points of Poisson-Nijenhuis structures.
method Completely describe pairs of compatible Poisson structures near singular points.
result Pairs of compatible Poisson structures near singular points are completely described.
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
The paper explores obstructions for symplectic Lie algebroids on surfaces.
problem Obstacles to the existence of symplectic Lie algebroids.
method Analysis through characteristic classes of symplectic Lie algebroids.
result Full obstructions for surfaces to carry symplectic Lie algebroids.
The paper solves a generalized Kähler Calabi-Yau problem for nondegenerate structures.
problem Formulating a Calabi-Yau conjecture in generalized Kähler geometry for nondegenerate Poisson structures.
method Defining Hamiltonian deformation spaces, establishing uniqueness, and using GIT framework.
result All solutions are hyper-Kähler metrics, and the flow evolves within the given class.
We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. Marcut for the study of the above problem in the case of a Poisson manifold, we establish the existence, under a condition, of a nondegenerate Poisson-Nijenhuis structure on …
We classify linear Nambu structures (which are generalized Poisson structures in Hamiltonian dynamics and which give rise to integrable differential forms and singular foliations), then give a linearization for Nambu structures anf integrable differential forms near a nondegenerate singular point.
Note on linearizing certain Nambu structures.
problem Linearizing Nambu structures of coorder 1.
method Showing linearizability with closed integrable differential form.
result Nambu structures of coorder 1 can always be linearized.
New type of manifolds derived from Poisson structures.
problem Generalizing Poisson Nijenhuis manifolds.
method Introducing pseudo-Poisson Nijenhuis manifolds and showing their properties.
result Found new materials to construct Courant algebroids.
We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie a…
The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.
problem Understanding the relationship between Poisson-Lie structures and invariant volume forms in Hamiltonian systems.
method Analyzing the existence and preservation of invariant volume forms under Hamiltonian vector fields on Poisson-Lie groups.
result A unimodular Poisson-Lie structure ensures the preservation of a multiple of any left-invariant volume, and the existence of a preserving volume form implies unimodularity.
In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homolo…
Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the fundamental group of a compact connected orientable topological surface with finitely m…
New ansatz for generalized Kähler surfaces derived from hyperKähler ansatz.
problem Deriving a new ansatz for generalized Kähler surfaces.
method Generalized Gibbons-Hawking ansatz for nondegenerate Poisson structure with biholomorphic S1 action. result Classification of all complete solutions with smallest symmetry group.
We show that aff(n), the Lie algebra of affine transformations of Rn, is formally and analytically nondegenerate in the sense of A. Weinstein. This means that every analytic (resp., formal) Poisson structure vanishing at a point with a linear part corresponding to aff(n) is locall…
In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general cas…
Study Frobenius pencils and compatible non-homogeneous Poisson structures.
problem Compatibility of multicomponent local Poisson structures.
method Algebraic interpretation via Frobenius algebras and classification of Frobenius pencils.
result Classification of Frobenius pencils under generic conditions.
Study on symmetries and equilibria in Poisson manifolds, with applications to rigid body dynamics.
problem Characterizing conformal relative equilibria on Poisson manifolds.
method Introducing conformally Poisson actions and momentum maps, establishing algebraic criteria.
result Classification of nontrivial conformal relative equilibria in Lie algebras, with applications to rigid body dynamics.
Simplified definition of LA-Courant algebroids and Poisson Lie 2-algebroids.
problem Defining and characterizing LA-Courant algebroids and Poisson Lie 2-algebroids.
method Using split Lie 2-algebroids and self-dual 2-representations to define LA-Courant algebroids, and studying geometric examples and induced structures.
result New examples of Poisson Lie 2-algebroids and a new construction of Courant algebroids.
We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a …
Contact structures are induced by nondegenerate skew fibrations of R^3.
problem Understanding contact structures induced by skew fibrations of R^3.
method Characterizing nondegenerate fibrations and using a recent result on tightness in contact metric 3-manifolds.
result The plane field induced by any nondegenerate fibration is a tight contact structure.
Study on CR structures in 7D, proving maximal symmetry dimension.
problem Proving symmetry dimension bound for CR structures in 7D.
method Investigating homogeneous models and proving uniqueness.
result 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in 7D.
Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.
problem Classify and understand nondegenerate fibrations of Euclidean spaces.
method Topological and geometric analysis of nondegenerate fibrations, including continuity at infinity.
result Prove that every germ of a nondegenerate fibration extends to a global fibration.
We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.
problem CR hypersurfaces in C^4 with constant rank Levi form and their defining equations.
method Obtained a complete normal form for models of real analytic uniformly 2-nondegenerate CR hypersurfaces in C^4.
result Found explicit formulas for infinitesimal symmetries of homogeneous 2-nondegenerate models.
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket…
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.
problem Characterizing and simplifying the defining equations of 2-nondegenerate CR hypersurfaces.
method Characterization of 2-nondegenerate models, derivation of normal forms, computation of CR invariants, derivation of infinitesimal symmetries.
result The moduli space of 2-nondegenerate CR hypersurfaces in C^N is infinite dimensional for N>3.
Characterizes CR manifolds in complex flag manifolds.
problem Closed real orbits in complex flag manifolds.
method Characterization through CR manifold structures and real forms.
result Closed orbits are finitely nondegenerate.
We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to …
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.
problem Classifying CR structures and their leaf spaces.
method Using unit tangent bundles and dynamical Legendrian contact structures.
result New examples of 2-nondegenerate CR structures are provided.
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.
Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair (Ω,ω), such that Ω is a symplectic form and ω is a 3-differential form which satisfies ω∧Ω=0 and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…
Defines pre-Kähler structures and their properties.
problem Geometry of Levi degenerate CR hypersurfaces.
method Introduces pre-Kähler structures, holomorphic degeneration, and finite-nondegeneracy.
result Symmetry algebra is finite-dimensional if and only if structure is finitely nondegenerate.
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
This paper constructs an explicit {e}-structure for certain 2-nondegenerate hypersurfaces.
problem Characterizing and classifying 2-nondegenerate Levi rank 1 hypersurfaces in complex space.
method Normalization of group parameters and construction of an explicit {e}-structure.
result An explicit {e}-structure is constructed for hypersurfaces where primary invariants do not vanish.
New method classifies symplectic structures on Lie groups.
problem Classifying left-invariant symplectic structures on Lie groups.
method Using moduli space of left-invariant nondegenerate 2-forms.
result Classified left-invariant symplectic structures on specific Lie groups.
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise i…
New metrics produce discrete zero sets for nondegenerate harmonic forms.
problem Creating metrics to produce discrete zero sets for nondegenerate harmonic forms.
method Metric perturbation to produce new nondegenerate harmonic forms with discrete zero sets.
result Existence of metrics producing discrete zero sets for nondegenerate harmonic forms.
New CR hypersurfaces in complex space with specific properties.
problem Constructing CR hypersurfaces with arbitrary nilpotent symbols.
method Introduced a class of CR hypersurfaces with methods applicable to all cases with N>5. result Solved equivalence problem for structures with a single Jordan block symbol.
Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…
Reformulates Fock-Rosly Poisson structure using quasi-triangular r-matrices.
problem Defining Fock-Rosly Poisson structure on moduli spaces.
method Using Lie algebra actions and quasi-triangular r-matrices.
result Shows Fock-Rosly structure as mixed product Poisson structure.
Global Poisson structures on S4 studied using twistor methods.
problem Global Poisson structures on S4. method Holomorphic Poisson structures on CP3, twistor method. result Examples of Poisson structures on S4 associated with codimension one holomorphic foliations of degree 2 on CP3. New Poisson structures on algebras linked to derivatives.
problem Understanding Poisson structures on trivial extension algebras.
method Introducing a correspondence between Poisson structures and derivatives, and formulating results in terms of Lie algebroids.
result One-to-one correspondence between Poisson structures and data involving derivatives.
Study Lie algebras with specific bilinear forms, finding exceptions.
problem Characterizing Lie algebras with symmetric, invariant, and nondegenerate bilinear forms.
method Analyzing structural properties and exceptions of Lie algebras.
result Rare exceptions to the properties of Lie algebras with these bilinear forms.
Investigates CR structures in 7D, showing 8 is max symmetry dimension.
problem CR structures in 7D with intransitive symmetry.
method Various methods to bound symmetry dimension, demonstrating existence of non-equivalent models.
result Existence of infinitely many non-equivalent submaximally symmetric models.