Search

We study holomorphic isometries between bounded symmetric domains with respect to the Bergman metrics up to a normalizing constant. In particular, we first consider a holomorphic isometry from the complex unit ball into an irreducible bounded symmetric domain with respect to the Bergman metrics. In this direction, we show that images of (nonempty) affine-linear sections of the complex unit ball must be the intersections of the image of the holomorphic isometry with certain affine-linear subspaces. We also construct a surjective holomorphic submersion from a certain subdomain of the target bounded symmetric domain onto the complex unit ball such that the image of the holomorphic isometry lies inside the subdomain and the holomorphic isometry is a global holomorphic section of the holomorphic submersion. This construction could be generalized to any holomorphic isometry between bounded symmetric domains with respect to the \emph{canonical K\"ahler metrics}. Using some classical results for complex-analytic subvarieties of Stein manifolds, we have obtained further geometric results for images of such holomorphic isometries.

We study local holomorphic maps between K\"ahler manifolds preserving $(p,p)$-forms. In this direction, we prove that any such local holomorphic map $F$ is a holomorphic isometry up to a scalar constant provided that $p$ is strictly less than the complex dimension of the domain of $F$. We then study local holomorphic maps between finite dimensional complex space forms preserving invariant $(p,p)$-forms. It was proved by Calabi that there does not exist a local holomorphic isometry between complex space forms $M$ and $N$ provided that $M$ and $N$ are of different types. In this article, we generalize this result to local holomorphic maps between complex space forms $M$ and $N$ preserving invariant $(p,p)$-forms whenever $M$ and $N$ are of different types except for the case where the universal covers of $M, N$ are biholomorphic to $\mathbb{C}^m, \mathbb{P}^n$, respectively and $2\le p=m

We study proper holomorphic maps between type-$\mathrm{I}$ irreducible bounded symmetric domains. In particular, we obtain rigidity results for such maps under certain assumptions. More precisely, let $f:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p',q'}$ be a proper holomorphic map, where $p\ge q\ge 2$ and $q'<\min\{2q-1,p\}$. Then, we show that $p'\ge p$ and $q'\ge q$. Moreover, we prove that there exist automorphisms $\psi$ and $\Phi$ of $D^{\mathrm{I}}_{p,q}$ and $D^{\mathrm{I}}_{p',q'}$ respectively, such that $f=\Phi\circ G_h\circ \psi$ for some map $G_h:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p',q'}$ defined by \[ G_h(Z):= \begin{bmatrix} Z & {\bf 0}\\ {\bf 0} & h(Z) \end{bmatrix}\quad \forall\; Z\in D^{\mathrm{I}}_{p,q},\] where $h:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p'-p,q'-q}$ is a holomorphic map., Comment: The proof of Proposition 6.6 in version 1 of the article is amended. This proposition becomes Proposition 6.8 in version 2 of the article

We study proper holomorphic maps between bounded symmetric domains $D$ and $\Omega$. In particular, when $D$ and $\Omega$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper holomorphic map from $D$ to $\Omega$ is a totally geodesic holomorphic isometric embedding with respect to certain canonical K\"ahler metrics of $D$ and $\Omega$. We also obtain some results regarding holomorphic maps $F:D\to \Omega$ which map minimal disks of $D$ properly into rank-$1$ characteristic symmetric subspaces of $\Omega$. On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between $D$ and $\Omega$ under a certain rank condition on $D$ and $\Omega$., Comment: To appear in Proceedings of the American Mathematical Society

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincar\'e disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $\Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincar\'e disk with ${\rm Aut}(\Omega')$-equivalent tangent spaces into a tube domain $\Omega' \subset \Omega$ and derive a contradiction by means of the Poincar\'e-Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z \subset \Omega$. More precisely, if $\check \Gamma\subset {\rm Aut}(\Omega)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z/\check \Gamma$ is compact, we prove that $Z \subset \Omega$ is totally geodesic. In particular, letting $\Gamma \subset{\rm Aut}(\Omega)$ be a torsion-free cocompact lattice, and $\pi: \Omega \to \Omega/\Gamma =: X_\Gamma$ be the uniformization map, a subvariety $Y \subset X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $\pi^{-1}(Y)$ is an algebraic subset of $\Omega$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_\Gamma$ by means of bi-algebraicity without recourse to the celebrated monodromy result of Andr\'e-Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices., Comment: v2: The proof of Theorem 5.23 is amended; v3: correction of typos and minor changes on wording

In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.

We first give an exposition on holomorphic isometries from the Poincar\'e disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.

We study general properties of holomorphic isometric embeddings of complex unit balls $\mathbb B^n$ into bounded symmetric domains of rank $\ge 2$. In the first part, we study holomorphic isometries from $(\mathbb B^n,kg_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ with non-minimal isometric constants $k$ for any irreducible bounded symmetric domain $\Omega$ of rank $\ge 2$, where $g_D$ denotes the canonical K\"ahler-Einstein metric on any irreducible bounded symmetric domain $D$ normalized so that minimal disks of $D$ are of constant Gaussian curvature $-2$. In particular, results concerning the upper bound of the dimension of isometrically embedded $\mathbb B^n$ in $\Omega$ and the structure of the images of such holomorphic isometries were obtained. In the second part, we study holomorphic isometries from $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ for any irreducible bounded symmetric domains $\Omega\Subset \mathbb C^N$ of rank equal to $2$ with $2N>N'+1$, where $N'$ is an integer such that $\iota:X_c\hookrightarrow \mathbb P^{N'}$ is the minimal embedding (i.e., the first canonical embedding) of the compact dual Hermitian symmetric space $X_c$ of $\Omega$. We completely classify images of all holomorphic isometries from $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ for $1\le n \le n_0(\Omega)$, where $n_0(\Omega):=2N-N'>1$. In particular, for $1\le n \le n_0(\Omega)-1$ we prove that any holomorphic isometry from $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ extends to some holomorphic isometry from $(\mathbb B^{n_0(\Omega)},g_{\mathbb B^{n_0(\Omega)}})$ to $(\Omega,g_\Omega)$.

We first study holomorphic isometries from the Poincar\'e disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain of rank $\ge 2$ which is not biholomorphic to any type-$\mathrm{IV}$ domain. In particular, our study provides many new examples of holomorphic isometries from the Poincar\'e disk into irreducible bounded symmetric domains of rank at least $2$ except for type-$\mathrm{IV}$ domains.