Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
We study holomorphic extensions of Matsuki orbits in complex Grassmannians.
problem Analyzing the analytic continuation of Matsuki orbits in complex Grassmannians.
method Using Rossi's theory of holomorphic extension and the holomorphic fiber bundle structure, we establish that the envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. result The envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. Explains complex adjoint orbits in Lie theory and geometry.
problem Understanding adjoint orbits in Lie theory and geometry.
method Expository introduction to adjoint orbits of complex semisimple groups.
result Provides insights into properties of semisimple and nilpotent orbits.
This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.
problem Investigating Hamiltonian systems and Kähler structures on complex coadjoint orbits.
method Analyzes (pseudo)-holomorphic Hamiltonian systems, Lefschetz and almost toric fibrations, and introduces pseudo-holomorphic Hamiltonian systems.
result Complex coadjoint orbits exhibit both Hyperkähler and holomorphic Kähler structures, suggesting Kähler duality.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
We investigate the CR geometry of the orbits M of a real form G0 of a complex simple group G in a complex flag manifold X=G/Q. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical G0-equivariant and Mostow fibrations, and topological properties of the orbits.
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.
The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as counterpart of a hyperpolar action on a symmetric space of compact type. In this paper, we construct examples of a complex hyperpolar action without singular orbit and investigate the geometry of the orbit…
Let Ĝ be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of Ĝ. The flag manifold Ĝ/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we charac…
Approximates symplectic automorphisms of coadjoint orbits using Hamiltonian Carleman methods.
problem Approximating symplectic automorphisms of coadjoint orbits.
method Hamiltonian Carleman approximation for coadjoint orbits of complex Lie groups.
result Established the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
Study Sp(n)-orbits in complex and Σ-complex subspaces of Hermitian quaternionic vector spaces.
problem Characterize Sp(n)-orbits in Grassmannians of complex and Σ-complex subspaces. method Decompose subspaces into 4-dimensional complex addends and 2-dimensional totally complex subspace. Use properties of isoclinic subspaces and principal angles.
result Determine full set of invariants for Sp(n)-orbits in GrR(2k,4n). This research shows that Mishchenko-Fomenko subalgebras are completely integrable on all regular orbits.
problem The complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits.
method The approach incorporates the theory of regular sl2-triples and associated Slodowy slices, as developed by Kostant. result Each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits.
Study exhaustions for complex orbits in almost homogeneous manifolds.
problem Complex orbits in almost homogeneous manifolds.
method Complex homogeneous Monge-Ampère equations.
result Rigidity results on complex spaces.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
We characterize isometric actions on compact Kaehler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of …
The paper calculates the size of origami orbit graphs in complex surfaces.
problem Calculating the size of origami orbit graphs in complex surfaces.
method Classification of SL(2,Z)-orbits of primitive origamis and reuse of machinery for Prym eigenforms. result Diameter bounds of O(N2/3logN) for orbit graphs in H(2) and H(4), H(6). The paper examines Hamiltonian stability of Lagrangian tori in complex hyperbolic spaces.
problem Investigating Hamiltonian stability of Lagrangian tori in complex hyperbolic spaces.
method Standard Hamiltonian Tn-action on CHn; proving stability and rigidity results. result Existence of infinitely many H-unstable Tn-orbits when n≥3. We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group G acting linearly and rationally on a real vector space V. G can be viewed as the real points of a complex reductive group GC which acts on $V…
We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.
Study infinitesimally tight Lagrangian orbits in symplectic manifolds.
problem Understanding Lagrangian orbits in symplectic manifolds.
method Analyzing isotropic orbits of Lie group actions with equivariant moment maps.
result Examples of Lagrangian orbits in complex flag manifolds and cotangent bundles.
The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
problem Classifying Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
method Examining orbits as CR submanifolds, proving total geodesy, and analyzing contact structures.
result Completely determine Sasaki-Einstein orbits.
We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to t…
We prove a relation between the ∂ˉM cohomology of a minimal orbit M of a real form G0 of a complex semisimple Lie group G in a flag manifold G/Q and the Dolbeault cohomology of the Matsuki dual open orbit X of the complexification K of a maximal compact subgroup K0 of G0, under the assum…
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.
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…
For a representation of a finite group G on a complex vector space V we determine when a holomorphic (qp)-tensor field on the principle stratum of the orbit space V/G can be lifted to a holomorphic G-invariant tensor field on V. This extends also to connections. As a consequence we determine those h…
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
Classifies finite orbits of mapping class group actions on surface group representations.
problem Finite orbits of mapping class group actions on surface group representations.
method Classification based on surface genus and representation properties.
result Finite orbits correspond to homomorphisms with finite image for genus at least two, and to finite or special dihedral representations for genus one.
Researchers create a new metric on complex projective space bundles.
problem Constructing a hyperkähler metric on complex projective space bundles.
method Explicit construction in local coordinates, using holomorphic isomorphism to coadjoint orbits.
result A hyperkähler metric on twisted cotangent bundles of CPn. New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
Geometric bijection and homotopy equivalence between Lie group orbits.
problem Isomorphic adjoint and coadjoint representations of Lie groups.
method Geometric bijection and homotopy equivalence of orbits.
result Geometrically defined bijection and homotopy equivalence between adjoint and coadjoint orbits.
New framework mated Kleinian groups with complex polynomials, revealing unique group properties.
problem Mating Kleinian groups with complex polynomials dynamics.
method Orbit equivalence framework for holomorphic mating, focusing on Fuchsian groups and higher Bowen-Series maps.
result Only torsion-free Fuchsian groups can be mated, with specific properties of Bowen-Series maps.
Let J1 be the real form of complex simple Jordan algebra with the automorphism group G of type F4(−20). Explicitly, we give the orbit decomposition of J1 under the action of G and determine the Lie group structure of stabilizer for each G-orbit on J1.
Study on almost Kaehler geometry of Lie groups orbits.
problem Understanding the geometry of adjoint orbits of Lie groups.
method Explicit formulas for Chern-Ricci form, scalar curvature, and Nijenhuis tensor derived from root data.
result Explicit formulas and conditions for the Chern-Ricci form and Kaehler type quotients.
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…
The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.
problem Characterizing Lagrangian submanifolds in adjoint semisimple orbits.
method Analyzing real flags and orbits of real forms with respect to symplectic forms.
result Classification of infinitesimally tight Lagrangian submanifolds in the compact case and Lagrangian submanifolds in the complex case.
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non-hR minimal closures. In this paper, we investigate a curvature-adapted and proper complex equifocal submanifold in a symmetric space of non-compact type. The class of these submanifolds contains principal orbits of Hermann type actions as homogeneous examples. In future, the results in this paper will be used to give a submanifold geometri…
Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
The study finds dense orbits and absolute period leaves for complex flows.
problem Existence of dense orbits for real Rel flows on holomorphic 1-forms.
method Established a density criterion for mSL(2,R)-orbit closures, verified using explicit constructions. result Found dense leaves and examples of absolute period foliation.
We classify polar actions on complex hyperbolic spaces up to orbit equivalence.
We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to any Riemannian metric. After S. Novikov's work and a complement by J.-C. Sikorav, under some genericity…
Deforms orbits in Lie algebras to Lagrangian submanifolds.
problem Deforming orbits in semisimple Lie algebras.
method Coadjoint orbit deformation and Hermitian symplectic form.
result Constructs Lagrangian submanifolds.
Proves properties of periodic billiard orbits in ellipses.
problem Understanding periodic orbits in ellipses.
method Geometric and complex analytic methods.
result Sum of cosines of angles remains constant in one-parameter family of polygons.
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…
Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.