arXiv research

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

We prove that if $(M^n,g)$, $n \ge 4$, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) $M$ admits a metric with positive isotropic curvature (ii) $(M,g)$ is isometric to a locally symmetric space (iii) $(M,g)$ is K…

Geometric proof confirms a conjecture about a metric on Cartan-Hartogs domains.

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.

Let $M^n$ be a compact Kähler manifold with bisectional curvature bounded from below by $1$. If $diam(M) = π/ \sqrt{2}$ and $vol(M)> vol(\mathbb{C}\mathbb{P}^n)/ 2^n$, we prove that $M$ is biholomorphically isometric to $\mathbb{C}\mathbb{P}^n$ with the standard Fubini-Study metric.

Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dime…

Study shows most isometric submersions between Teichmüller spaces are forgetful.

Let $(M^n, g)$ be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold $N^k$ with $c_1 < 0$. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive b…

Proper holomorphic isometries between Bergman domains are biholomorphisms.

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…

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a nat…

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

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal…

Complex domains covering manifolds are biholomorphic to balls.

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

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

We prove that the Teichmüller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.

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

The paper proves global invertibility for certain 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 …

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.

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

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…

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.

Researchers create normal forms for CR manifolds in complex space.

Study complex structures of hyperkähler manifolds with infinite type.

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.

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

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.

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 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.

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

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

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…

The paper describes complex structures on Oeljeklaus-Toma manifolds.

Holomorphic actions on complex spaces for nilpotent groups.