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 study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Characterizes projective special complex manifolds using c-projective structures.
problem Characterizing projective special complex manifolds.
method Defining S1-bundles and constructing conical special complex manifolds. result Intrinsic characterization of projective special complex manifolds.
In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák's theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert's Fourth Problem is also pointed out. As an…
The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.
Killing tensors on complex projective space are identified and generated by Killing fields.
problem Identifying Killing tensors on complex projective space.
method Determining Killing tensors of arbitrary rank on complex projective space with Fubini-Study metric.
result Complex projective spaces are generated by Killing fields.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Classifies foliations of complex and quaternionic projective spaces.
problem Classifying isoparametric foliations of complex and quaternionic projective spaces.
method Investigating projections of inhomogeneous isoparametric foliations of the 31-sphere under Hopf fibrations.
result Solved the last remaining open cases in the classification.
Extremal Kähler submanifolds of complex projective spaces have natural extensions.
problem Understanding the structure and properties of extremal Kähler submanifolds.
method Analyzing the natural extensions and holomorphic isometric immersions of extremal Kähler submanifolds.
result Every connected extremal Kähler submanifold has a natural extension which is a complete Kähler manifold.
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
Estimates Schwarzian derivative on long complex projective tubes.
problem Behaviour of Schwarzian derivative on complex projective structures.
method Analyzes Schwarzian derivative on long complex projective tubes, estimating its pairing with infinitesimal earthquakes and graftings.
result Obtains bounds for the variation of renormalized volume under complex earthquake paths and its asymptotic behavior under pinching.
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. We simplify the construction of projection complexes due to Bestvina-Bromberg-Fujiwara. To do so, we introduce a sharper version of the Behrstock inequality, and show that it can always be enforced. Furthermore, we use the new setup to prove acylindricity results for the action on the projection complexes. We also trea…
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…
We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Furthermore we show that the (pseudo-)Kähler metrics defined on some domain in the projective plane which are compatible with the standard complex projective struct…
Solves classical problem with Kähler-Einstein metrics in complex projective spaces.
problem Classical problem of non-isometric bidimensional Kähler-Einstein submanifolds.
method Listed complete non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds.
result Solves the classical problem in the specified case.
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly Kähl…
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
Listed Kaehler-Einstein manifolds in complex projective spaces.
problem Classical problem of finding non-isometric Kaehler-Einstein manifolds.
method Complete list for n≤6 using Tn-invariant manifolds. result Solved the problem for n≤6. We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.
A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.
Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology.
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.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of comple…
The study classifies stable submanifolds in product spaces of projective spaces.
problem Classifying stable submanifolds in product spaces of projective spaces.
method Provided a classification theorem for compact stable minimal immersions in product spaces of projective spaces.
result Characterized complex minimal immersions in the product of two complex projective spaces.
The paper finds transformation formulas for quaternionic complex structures.
problem Quaternionic projective invariance of k-Cauchy-Fueter complex. method Explicit transformation formulae under mSL(n+1,H). result Quaternionic projectively invariant operator and defining density.
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
Determines higher smooth surgery structure sets of complex projective spaces.
problem Understanding the higher smooth surgery structure sets of complex projective spaces.
method Analyzes the free subgroup and torsion in low dimensions.
result Obtains information in all dimensions for the free subgroup.
Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of Pn of codimension one. As a consequence …
In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler ma…
Max diameter Kahler manifolds with positive bisectional curvature are complex projective spaces.
problem Understanding the rigidity of Kahler manifolds with specific curvature properties.
method Proving isometry to complex projective space using maximal diameter and positive bisectional curvature.
result Kahler manifolds with maximal diameter and positive bisectional curvature are isometric to complex projective spaces.
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.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy ρ:π1(S)→ PSL$_2\mathbb…
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. We describe the fundamental groups of ordered and unordered k point sets in complex projective space of dimension n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of points.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
We extend T. Y. Thomas's approach to the projective structures, over the complex analytic category, by involving the ρ-connections. This way, a better control of the projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold P …
New characterizations of ruled real hypersurfaces in complex projective space found.
problem Characterizing ruled real hypersurfaces in complex projective space.
method Defined tensor fields related to Levi-Civita and generalized Tanaka-Webster connections and studied the structure operator.
result Obtained new characterizations of ruled real hypersurfaces in complex projective space.
Flexible links have symplectic representatives in complex projective space.
problem Existence of symplectic representatives for flexible links.
method Construction of a symplectic surface invariant under complex conjugation.
result No obstructions to finding symplectic representatives beyond classical topology.
Close to complex projective spaces, Ricci shrinkers are rigid.
problem Rigidity of complex projective spaces in Ricci shrinkers.
method Proving isometry using Gromov-Hausdorff distance.
result Ricci shrinkers close to (CPN,gFS) are isometric to (CPN,gFS). An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.