Normalizes submanifolds near Levi-nondegenerate hyperquadrics.
problem Normalizing submanifolds near Levi-nondegenerate hyperquadrics.
method Adapting techniques for d=1 to prove Chern-Moser normal form theorem.
result New direct proof of Chern-Moser normal form theorem.
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
Biholomorphisms of transport twistor spaces are rigid under certain conditions.
problem Rigidity of biholomorphisms in transport twistor spaces.
method Proof of rigidity for biholomorphisms between transport twistor spaces of simple or Anosov surfaces.
result Biholomorphisms are rigid, up to constant rescaling and the antipodal map, being lifts of orientation-preserving isometries.
Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
Complex domains covering manifolds are biholomorphic to balls.
problem Covering compact manifolds by bounded domains with smooth boundaries.
method Using biholomorphic mappings and properties of C1,1 boundaries. result Bounded domains with smooth boundaries covering compact manifolds are biholomorphic to the unit ball.
Proves a complex structure conjecture for a specific type of Lie groups.
problem Proving a conjecture about left-invariant complex structures on nilpotent Lie groups.
method Analyzes simply connected, nilpotent Lie groups of dimension 2n.
result Proves biholomorphism to C^n for the specified Lie groups.
Study extends biholomorphisms between convex domains in complex space without boundary constraints.
problem Extending biholomorphisms between convex domains without boundary regularity.
method Combining coarse geometry techniques with dynamical properties of maps in Gromov hyperbolic spaces.
result Proves extensions for biholomorphisms and quasi-isometries between convex domains.
Extends Carathéodory's theorem to multidimensional domains with constant curvature.
problem Characterizing biholomorphic domains with constant holomorphic curvature.
method Using Bergman representative coordinates and Calabi's diastasis.
result Provides sufficient conditions for the boundary of a biholomorphic ball to be a topological sphere.
The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.
problem Global invertibility of local diffeomorphisms and biholomorphisms in higher dimensions.
method The approach uses conformal geometry, complex analysis, elliptic PDEs, and topology.
result The main theorem guarantees global invertibility for specific local diffeomorphisms and biholomorphisms in higher dimensions.
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …
Holomorphic motions can't map to complex domains.
problem Characterizing mappings between holomorphic motions and complex domains.
method Analyzing biholomorphic properties of graph mappings.
result Graphs of holomorphic motions cannot be biholomorphic to strongly pseudoconvex domains.
Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.
problem Solving a 1975 problem posed by S. Kobayashi about complex Finsler vector bundles with positive Kobayashi curvature.
method Analyzes properties of complex Finsler vector bundles with positive Kobayashi curvature.
result Complex Finsler vector bundles with positive Kobayashi curvature are ample and biholomorphic to projective space.
Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.
problem Characterizing biholomorphic mappings between tube domains and bounded symmetric domains.
method Analyzing properties of Finsler symmetric cones and unital JB-algebras.
result Tube domains over Finsler symmetric cones are biholomorphic to bounded symmetric domains.
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
Study shows no global symmetries for Kähler metrics on compact manifolds.
problem Identifying symmetries in the space of Kähler metrics on compact manifolds.
method Proving isometries are induced by biholomorphisms and anti-biholomorphisms, analyzing Mabuchi's metric and its completion.
result No global symmetries exist for Kähler metrics on compact manifolds, including for Mabuchi's metric.
We introduce and study the notion of a biholomorphic gerbe with connection. The biholomorphic gerbe provides a natural geometrical framework for generalized Kahler geometry in a manner analogous to the way a holomorphic line bundle is related to Kahler geometry. The relation between the gerbe and the generalized Kahler…
A complex surface's Teichmüller space can't be locally convex.
problem Proving Teichmüller space's non-convexity locally.
method Analyzing properties of Teichmüller spaces and convex domains.
result Teichmüller space cannot be locally strictly convex.
Let (M,g) be a simply connected complete Kahler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in C^n, where dim M = n.
A unique Kähler potential on the unit ball is identified with constant differential norm.
problem Finding a unique Kähler potential with constant differential norm on the unit ball.
method Analyzing the Kähler potential of the unit ball and its biholomorphic equivalence to the Siegel domain.
result The Kähler potential of the Siegel domain is unique up to automorphisms with constant differential norms.
The study finds surfaces with specific curvature properties are essentially known manifolds.
problem Investigating curvature properties on Kähler manifolds.
method Analyzing the curvature operator of the second kind on closed Kähler surfaces.
result Closed Kähler surfaces with six-positive curvature operator of the second kind are biholomorphic to CP2. Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop) complex tangency, which are formally but not holomorphically normalizable (because…
Researchers create normal forms for CR manifolds in complex space.
problem Classifying and understanding 5D CR manifolds in C^4.
method Equivariant moving frames method to construct convergent normal forms.
result Complete normal forms for 5D CR submanifolds of C^4.
Study complex structures of hyperkähler manifolds with infinite type.
problem Understanding complex structures of hyperkähler four-manifolds with infinite topological type.
method Analyzing the Gibbons-Hawking ansatz and proving biholomorphic relationships to hypersurfaces or minimal resolutions of singular surfaces.
result For almost all complex structures, the manifold is biholomorphic to a hypersurface in \(\mathbb{C}^3\). For the remaining, it is biholomorphic to the minimal resolution of a singular surface under certain conditions.
We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g beco…
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…
Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
The Siu-Yang conjecture is reviewed for complex 2-manifolds.
problem Determining biholomorphic quotients of complex 2-balls.
method Review and analysis of existing results.
result Negative sectional curvature implies biholomorphic quotients of complex 2-balls.
Holomorphic maps of degree one are biholomorphic, confirming a partial order.
problem Understanding the partial order of holomorphic maps.
method Analyzing holomorphic maps of positive degree between compact complex manifolds.
result Holomorphic maps of degree one are biholomorphic.
Uniqueness proven for a specific type of complex manifold's solitons.
problem Proving uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons.
method Using a method to show uniqueness of the soliton vector field, which can be applied more widely.
result A noncompact complex manifold admits only one such soliton.
We prove that a complete noncompact Kähler manifold Mnof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}n and we show that the manifold is topologically {\bf R}2n. In particular, when Mn is a Kähler surface of positive bisecti…
Let X be a Stein manifold of complex dimension at least two, F:X→Cn a local biholomorphism, and q∈F(X). In this paper we formulate sufficient conditions involving only objects naturally associated to q, in order for the fiber over q to be finite. Assume that F−1(l) is 1-connec…
Applying a well known result for attracting fixed points of biholomorphisms \cite{RR, V}, we observe that one immediately obtains the following result: if (Mn,g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maxim…
The Kähler-Ricci flow converges to a negative Kähler-Einstein metric under certain conditions.
problem Regularity of long-time solutions to the Kähler-Ricci flow on compact manifolds.
method Parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics.
result The Ricci curvature is uniformly bounded on compact subsets away from singular fibers when generic fibers are biholomorphic.
Compact Kähler manifolds with specific curvature properties are rigid.
problem Diameter rigidity of Kähler manifolds with positive bisectional curvature.
method Proof of biholomorphic isometry under specific diameter and volume conditions.
result Compact Kähler manifolds with the specified properties are biholomorphically isometric to CPn. Study on LCK metrics and their existence criteria on complex manifolds.
problem Existence of LCK metrics on complex manifolds.
method Investigation of holomorphic torus actions and biholomorphisms on LCK type manifolds.
result Existence of Vaisman and LCK metrics under specific conditions on the manifold's biholomorphism group.
Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
problem Holonomy map properties for parabolic projective structures on non-compact surfaces.
method Proving the holonomy map is a local biholomorphism.
result Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
Let $\1$ and $\2$ be $\s$ domains in $\Cn$ and $f: \1 \rt \2$ an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C1 map to $ \bar \om_1$. We then prove that $f|_{\partial \1}: \partial \1 \rt \partial \2$ is a CR or anti-CR diffeomorphism. It follows that $\1$ and $\2$ must be bihol…
We study the equivalence problem for 4-dimensional CR-manifolds of CR-dimension 1 and codimension 2 which are referred to as Engel CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds through Cartan equivalence's method. In particular, we give the explicit expression of 4 biholomorphic …
Survey on hypothetical complex structure on 6-sphere.
problem Understanding the algebraic dimension and biholomorphisms of a hypothetical complex 6-sphere.
method Discussion of existing results and examples.
result Overview of Peternell--Campana--Demailly's result on algebraic dimension and Huckleberry--Kebekus--Peternell's on biholomorphisms.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
The paper describes complex structures on Oeljeklaus-Toma manifolds.
problem Investigating different complex structures on Oeljeklaus-Toma manifolds.
method Algebraic description of biholomorphic manifolds obtained by conjugating complex embeddings.
result Compact non-Kähler manifolds with multiple rigid complex structures.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.
The paper proves a special case of Yau's conjecture for Kähler surfaces.
problem Uniformization of complete noncompact Kähler surfaces with positive sectional curvature.
method Proves a complex Monge-Ampère equation to construct a plurisubharmonic weight function.
result A complete noncompact Kähler surface with positive and bounded sectional curvature is biholomorphic to \(\mathbb{C}^2\).
Study of conformal limits in Nakajima quiver varieties.
problem Understanding the conformal limits of Nakajima quiver varieties.
method Defined and studied a conformal limit construction for Nakajima quiver varieties, proving it is a limit of a one-parameter family and gives a biholomorphic map.
result Proved the conformal limit is a biholomorphic map between Lagrangian submanifolds of different quiver varieties.
Researchers study curvature invariants on complex domains, finding rigidity for unit balls.
problem Determine if obstruction flat boundary in C2 implies biholomorphic equivalence to the unit ball. method Analyze curvature invariants and their vanishing orders on bounded strictly pseudoconvex domains.
result The unit ball in C2 is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent …
Two new boundary Schwarz lemmas for holomorphic maps established.
problem Establishing boundary Schwarz lemmas for holomorphic maps.
method Two boundary versions of the Schwarz lemma for holomorphic maps.
result First boundary Schwarz lemmas for general holomorphic self maps and biholomorphisms with invariant Kähler metrics.