Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
Solves metrisability for 3D projective structures.
problem Metrisability of three-dimensional projective structures.
method Generic three-dimensional projective structures.
result Solved metrisability problem.
Extends projective structures using ρ-connections for better control.
problem Control projective flatness in complex analytic category.
method Involves ρ-connections to extend projective structures.
result Local identification of quaternionic projective space.
Two projective structures on Riemann surfaces are described and shown not to be identical.
problem Characterizing and distinguishing projective structures on Riemann surfaces.
method Construction and analysis of projective structures p u p_u p u and p h p_h p h using the period map and moduli space. result The Hodge projective structure p h p_h p h is not the same as the uniformization projective structure p u p_u p u . Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
Characterizes projective special complex manifolds using c-projective structures.
problem Characterizing projective special complex manifolds.
method Defining S 1 S^1 S 1 -bundles and constructing conical special complex manifolds. result Intrinsic characterization of projective special complex manifolds.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Maximal automorphisms found for complex projective structures.
problem Maximizing automorphisms in complex projective structures.
method Analyzing Fuchsian uniformizations and Galois Belyi curves.
result Fuchsian uniformizations of Hurwitz surfaces achieve maximal automorphisms.
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
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.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
Study projective and direct limits of Banach structures with connections to G G G -structures.
problem Understanding connections between Banach structures and G G G -structures. method Endow projective and direct limits with Fréchet or convenient structures and study connections.
result Illustrated examples demonstrate the study of projective and direct limits.
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's constructi…
A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From thi…
Complex projective structures can be joined via simple bubbling and debubbling.
problem Joining complex projective structures with quasi-Fuchsian holonomy.
method Performing (de)grafting via a sequence of one bubbling and one debubbling.
result Any complex projective structure can be joined to the uniformizing structure by a simple sequence of one bubbling and one debubbling.
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.
We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this con…
Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
Projective geometry simplifies Sasaki-Einstein structures and their compactification.
problem Understanding and compactifying Sasaki-Einstein structures.
method Projective differential geometry and holonomy reductions.
result Characterization of Sasaki-Einstein structures and their compactification.
The paper describes projective structures with torsion using formal frames and Thomas-Whitehead connections.
problem Characterizing projective structures with torsion.
method Using formal frames and Thomas-Whitehead connections.
result Fundamental theorem for Thomas-Whitehead connections regardless of torsion triviality.
Study how convex structures on surfaces change.
problem Degeneration of convex structures on surfaces.
method Analyzing bordered surfaces for convex projective structures.
result Understanding how convex structures degenerate.
New Einstein metrics linked to projective structures and Toda equations.
problem Linking projective structures to Toda equations and Einstein metrics.
method Established correspondence between projective structures and Toda equations, constructed new solutions, and linked to Einstein metrics.
result Explicit solutions of the S U ( ∞ ) SU(\infty) S U ( ∞ ) Toda equation and construction of Einstein metrics. The article constructs manifolds without real projective structure.
problem Finding manifolds without real projective structures.
method Generalizing Cooper and Goldman's example to manifolds with infinite fundamental group.
result Constructs a manifold with infinite fundamental group that does not admit a real projective structure.
In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.
We construct an infinite sequence of projectively flat manifolds by using castling transformations of prehomogeneous vector spaces. We also give a classification of manifolds equipped with a flat projective structure obtained by a finite number of castling transformations, and describe these flat projective structures …
Researchers use symplectic Dirac operator to study projective structure and its symmetries.
problem Understanding symmetries in projective structure and spinor fields.
method Realized symplectic spinor fields and operator in homogeneous projective structure framework.
result Symplectic Dirac operator's symmetry group is S L ~ ( 3 , R ) \widetilde{\mathrm{SL}}(3,{\mathbb R}) SL ( 3 , R ) . By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…
The paper proves a grafting theorem for meromorphic projective structures and shows the monodromy map is a local homeomorphism.
problem Understanding projective structures on Riemann surfaces with poles.
method Proves a grafting theorem involving crowned hyperbolic surfaces and uses the monodromy map to a decorated character variety.
result The monodromy map to the decorated character variety is a local homeomorphism.
Study identifies subvarieties of projective varieties mapping to models.
problem Understanding mappings of subvarieties to models on projective varieties.
method Analyzes smooth projective varieties with holomorphic locally homogeneous structures.
result Determines all subvarieties mapping to the model.
A new projective structure on Riemann surfaces differs from the uniformization theorem's structure.
problem Comparing two projective structures on compact Riemann surfaces.
method Using Hodge theory and the Torelli map to compare the ( 0 , 1 ) (0,1) ( 0 , 1 ) -component of the differential of sections of moduli spaces. result The two projective structures differ in general, with the ( 0 , 1 ) (0,1) ( 0 , 1 ) -component of the differential of the section corresponding to η ^ \widehat{\eta} η being a nonzero constant multiple of the Siegel form. The paper surveys partial results on convex real projective orbifolds with specific ends.
problem Characterizing convex real projective orbifolds with radial or totally geodesic ends.
method Proving homeomorphisms between deformation spaces and strata of projective structures.
result A homeomorphism between deformation spaces of convex real projective structures on orbifolds with radial or totally geodesic ends.
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
Properness proven for circle packings and Delaunay patterns on complex projective structures.
problem Proving properness for circle packings and Delaunay patterns on complex projective structures.
method Considering circle packings and Delaunay circle patterns on surfaces with complex projective structures, proving properness of the forgetful map.
result Proved properness of the forgetful map sending circle packings and Delaunay patterns to underlying complex structures.
New framework generalizes complex projective structures, proving non-existence of certain structures on Calabi-Yau manifolds.
problem Proving non-existence of certain holomorphic structures on Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries, proving independence of results, and using vector bundle properties.
result Non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures.
Notes on Fock and Goncharov's moduli spaces of real projective structures.
problem Describing moduli spaces of real projective structures on surfaces.
method Using Fock and Goncharov's methods and deducing results from their work.
result Results of Marquis and Goldman as consequences of their work.
The paper studies conformal and projective structures using 2-frame bundles.
problem Understanding connections between conformal and projective structures.
method Using the dressing field method to obtain local, gauge invariant connections.
result A projective tractor bundle can be defined using the same construction as for conformal structures.
Study on entropy of bulging deformations of projective structures on surfaces.
problem Entropy of bulging deformations in real projective structures on surfaces.
method Analysis of topological entropy in terms of bulging deformations.
result Construction of divergent structures with convergent topological entropy.
Study real projective structures on a specific Coxeter orbifold.
problem Characterize real projective structures on a noncompact Coxeter orbifold.
method Embedding and extending a Coxeter quadrilateral, perturbing to form a convex polytope, and analyzing the deformation space.
result Determine the detailed properties of the deformation space of real projective structures on the orbifold.
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
The paper explores projective structures on curves and their applications in conformal geometry.
problem Finding qualitative information about solutions of Hill equations.
method Detailed description of projective structures and their isomorphism classes, correcting previous inaccuracies.
result The Yamabe problem for curves has no general solutions in a conformal/Möbius ambient space.
Study projective structures and rational curves to understand Painlevé equations.
problem Analyzing projective structures and rational curves on surfaces.
method Analytic classification, normal forms, pencil/fibration decomposition, infinitesimal symmetries.
result Deduced transcendental results about Painlevé equations.