We study general properties of images of holomorphic isometric embeddings of complex unit balls ${\mathbb {B}}^m$ into irreducible bounded symmetric domains ${\varOmega }$ of rank at least 2 In particular, we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections ${\varLambda }$ of the compact dual $X_c$ of ${\varOmega }$ The question naturally arises as to which linear sections $Z = {\varLambda }\cap {\varOmega }$ are actually images of holomorphic isometries of complex unit balls We study the latter question in the case of bounded symmetric domains ${\varOmega }$ of type IV, alias Lie balls, ie, bounded symmetric domains dual to hyperquadrics We completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains ${\varOmega }$ Especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by Mok (Proc Am Math Soc 144:4515–4525, 2016) from varieties of minimal rational tangents, and that moreover any holomorphic isometric embedding $f: {\mathbb {B}}^m \rightarrow {\varOmega }$ extends to a holomorphic isometric embedding $f: \mathbb B^{n-1} \rightarrow {\varOmega }$, $\dim {\varOmega }= n$ The case of Lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type-IV domain are expected to be more rigid

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Holomorphic isometries of $${\mathbb {B}}^m$$ into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals

The Arveson-Douglas Conjecture is a conjecture about essential normality of submodules, and quotient modules of some analytic Hilbert modules on polynomial rings. It originated from multi-variable operator theory but turns out to have various connections outside this field. In this survey, we will introduce the conjecture, describe its backgrounds, applications, and recent developments. Several related questions are also included in this survey.

A Survey on the Arveson-Douglas Conjecture

We study the regularity and algebraicity for mappings into classical domains. Among other things, we establish everywhere regularity and algebraicity results for 
$$C^2$$

 CR maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large.

Regularity of mappings into classical domains

In this article, we study local holomorphic conformal embeddings from B into B1 × · · · ×BNm with respect to the normalized Bergman metrics. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.

Rigidity for local holomorphic conformal embeddings from B_n into B_{N_1}×…× B_{N_m}.

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 $(\Omega,g_\Omega)$.

On the structure of holomorphic isometric embeddings of complex unit balls into bounded symmetric domains

We construct isometric holomorphic embeddings of the unit ball into higher rank symmetric domains, first discovered by Mok, in an explicit way using Jordan triple systems, and prove uniqueness results for all domains, including the exceptional domains of dimension 16 and 27.

Holomorphic isometries from the unit ball into symmetric domains

Dixmier trace for Toeplitz operators on symmetric domains

We construct rational isometric holomorphic embeddings of the unit ball into higher rank symmetric domains D, first discovered by Mok, in an explicit way using Jordan triple systems, and we classify all isometric embeddings into tube domains of rank 2. For symmetric domains of arbitrary rank, including the exceptional domains of dimension 16 and 27, respectively, we characterize the Mok type mapping in terms of a vanishing condition on the second fundamental form of the image of F in D.

Holomorphic Isometries from the Unit Ball into Symmetric Domains

We give natural constructions of number rigid determinantal point processes on the unit disc $\mathbb{D}$ with sub-Bergman kernels of the form \[ K_\Lambda(z, w) = \sum_{n\in \Lambda}(n+1) z^n \bar{w}^n, \quad z, w \in \mathbb{D}, \] with $\Lambda$ an infinite subset of the set of non-negative integers. Our constructions are given both in a deterministic method and a probabilisitc method. In the deterministic method, our proofs involve the classical Bloch functions.

Rigidity of determinantal point processes on the unit disc with sub-Bergman kernels

For Toeplitz operators on bounded symmetric domains of arbitrary rank, we define a Hilbert quotient module corresponding to partitions of length $1$ and prove that it belongs to the Macaev class ${\mathcal{L}}^{n,\infty}$. We next obtain an explicit formula for the Dixmier trace of Toeplitz commutators in terms of the underlying boundary geometry.

Dixmier Trace for Toeplitz Operators on Symmetric Domains

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