arXiv research

Proper holomorphic isometries between Bergman domains are biholomorphisms.

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

Study extends biholomorphisms between convex domains in complex space without boundary constraints.

Study shows no global symmetries for Kähler metrics on compact manifolds.

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 $C^1$ 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 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 …

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…

Compact Kähler manifolds with specific curvature properties are rigid.

Holomorphic isometries from Poincaré disk to complex symmetric domains found.

We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension n, there exists a constant $Λ$, explicitly computable, such that any $Λ$-pinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.

Linear invariants of complex manifolds preserved by biholomorphisms.

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real param…

Proves conformal equivalence of visual metrics in smooth pseudoconvex domains.

We construct a Kähler structure (${\mathbb{J}},Ω,{\mathbb{G}}$) on the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space ${\mathbb{H}}^3$ and investigate its properties. We prove that (${\mathbb{L}}({\mathbb{H}}^3),{\mathbb{J}})$ is biholomorphic to ${\mathbb{P}}^1\times{\mathbb{P}}^1-\ba…

Local holomorphic maps preserving (p,p) forms are shown to be isometries.

We determine an explicit expression for the Ricci tensor of a K-manifold, that is of a compact Kaehler manifold M with vanishing first Betti number, on which a semisimple group G of biholomorphic isometries acts with an orbit of codimension one. We also prove that the Kaehler form and the Ricci form of M are uniquely d…

Complex domains covering manifolds are biholomorphic to balls.

Proves a complex structure conjecture for a specific type of Lie groups.

Extends Carathéodory's theorem to multidimensional domains with constant curvature.

The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

Study shows most isometric submersions between Teichmüller spaces are forgetful.

Holomorphic motions can't map to complex domains.

Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.

Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.

The paper proves bounded domains are biholomorphic to the unit ball.

For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be unique. By means of such connection, an absolute parallelism on U is determined and a…

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.

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.

The study finds surfaces with specific curvature properties are essentially known manifolds.

Researchers create normal forms for CR manifolds in complex space.

Study complex structures of hyperkähler manifolds with infinite type.

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.

The Siu-Yang conjecture is reviewed for complex 2-manifolds.

Holomorphic maps of degree one are biholomorphic, confirming a partial order.

Uniqueness proven for a specific type of complex manifold's solitons.

We prove that a complete noncompact Kähler manifold $M^{n}$of 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 $M^{n}$ is a Kähler surface of positive bisecti…

Let $X$ be a Stein manifold of complex dimension at least two, $F : X \rightarrow \mathbb{C}^n$ a local biholomorphism, and $q \in F(X)$. 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 $(M^n,g)$ 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.

Study on LCK metrics and their existence criteria on complex manifolds.

Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to…

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 …

Study on when Bergman metrics of domains are induced by balls.