Holomorphic Isometric Embeddings Research Articles

On CR maps between hyperquadrics and Winkelmann hypersurfaces

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel–Son and a recent result of Huang–Lu–Tang–Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain Kähler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative.

Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

Holomorphic isometric embeddings of the complex two-plane Grassmannian into quadrics

The present article studies holomorphic isometric embeddings of the complex two-plane Grassmannian into quadrics. We discuss the moduli space of these embeddings up to gauge and image equivalence using a generalisation of do Carmo–Wallach theory.

Holomorphic isometric embeddings of the projective space into quadrics

Holomorphic isometric embeddings of the projective space into quadrics

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré 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 Poincaré disk with $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces into a tube domain $\Omega^\prime \subset \Omega$ and derive a contradiction by means of the Poincaré–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 $\check{\Gamma} \subset \operatorname{Aut} (\Omega)$ 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 $\Gamma \subset \operatorname{Aut} (\Omega)$ 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 $\pi-(Y)$ 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 André–Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of André–Deligne we exploit the existence theorem of Aubin and Yau on Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ and make use of Nadel’s semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

Boundary characterization of holomorphic isometric embeddings between indefinite hyperbolic spaces

We provide in this paper a boundary characterization in terms of the boundary CR invariants (or pseudo-Hermitian invariants) for holomorphic isometric embeddings between indefinite hyperbolic spaces of general codimensions.

On proper holomorphic maps between bounded symmetric domains

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

Remarks on holomorphic isometric embeddings between bounded symmetric domains

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.

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

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 $(\Omega,g_\Omega)$.

Holomorphic isometric embeddings of the projective line into quadrics

We discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

Classification Problem of Holomorphic Isometries of the Unit Disk Into Polydisks

We study the classification problem of holomorphic isometric embeddings of the unit disk into polydisks as in [Ng10, Ch16a]. We give a complete classification of all such holomorphic isometries when the target is the 4-disk Δ4. Moreover, we classify those holomorphic isometric embeddings with certain prescribed sheeting numbers. In addition, we prove that a known example in the space HIk(Δ,Δqk;q) is globally rigid for any integers k,q≥2, which generalizes Theorem 1.1 in [Ch16a].

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

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, i.e., 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.

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

We study the extension problem for germs of holomorphic isometries f: (D;x_0) \to (\Omega;f(x_0)) up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics ds_D^2 on D and ds_\Omega^2 on \Omega . Our main focus is on boundary extension for pairs of bounded domains (D,\Omega) such that the Bergman kernel K_D(z,w) extends meromorphically in (z,\overline w) to a neighborhood of \overline D \times D , and such that the analogous statement holds true for the Bergman kernel K_{\Omega}(\zeta,\xi) on \Omega . Assuming that (D;ds_D^2) and (\Omega;ds_\Omega^2) are complete Kähler manifolds, we prove that the germ of map f extends to a proper holomorphic isometric embedding such that Graph (f) extends to a complex-analytic subvariety on some neigborhood of \overline D \times \overline{\Omega} . In the event that the Bergman kernel K_D(z,w) extends to a rational function in (z;\overline w) and the analogue holds true for the Bergman kernel K_{\Omega}(\zeta,\xi) , we show that Graph (f) extends to an affine-algebraic variety. Our results apply especially to pairs (D,\Omega) of bounded symmetric domains in their Harish-Chandra realizations. When D is the complex unit ball B^n of dimension n \ge 2 , we obtain a new rigidity result which guarantees the total geodesy of the map under certain conditions. On the other hand, we construct examples of holomorphic isometries of the unit disk into polydisks which are not totally geodesic, answering in the negative a conjecture of Clozel-Ullmo's.

Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. 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 and 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.

On holomorphic isometric embeddings of the unit disk into polydisks

We study the classification of holomorphic isometric embeddings of the unit disk into polydisks. As a corollary of our results, we can give a complete classification when the target is the 2-disk and the 3-disk. We also prove that the holomorphic isometric embeddings between polydisks are induced by those of the unit disk into polydisks.

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F1(z) = (0, In;m(z)) with the isometric constant equal to (m + 1)/(n + 1) and F2(z) = (In;m(z), In;m(z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here \({I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}\) is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.

Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry f: (Δ, λ dsΔ2;0) → (Ω, dsΩ2;0) of the Poincare disk Δ into a bounded symmetric domain Ω ⋐ ℂN in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding F: (Δ, λ dsΔ2;) → (Ω, dsΩ2) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂN. Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρp: H → Hp and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H3 of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|2. On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincare disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t2u(τ), where u|t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt|t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincare disk into the polydisk must develop singularities along the boundary circle. The equation δu/δt|t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.

On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains

In this article we study holomorphic isometries of the Poincare disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk Δ into an irreducible bounded symmetric domain Ω ⋐ ℂN in its Harish-Chandra realization must extend to an affine-algebraic subvariety V C ⊂ = ℂ×ℂN = ℂN+1, and that the irreducible component of V∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F: Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δΔ. Starting with the structural equation for holomorphic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form a of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ∥ρ∥ must vanish at a general boundary point either to the order 1 or to the order ½ called a holomorphic isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

Inequalities of eigenvalues for the Dirac operator on compact complex spin submanifolds in complex projective spaces

For a compact complex spin manifold M with a holomorphic isometric embedding into the complex projective space, the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator, which depend on the data of an isometric embedding of M. Further, from the inequalities of eigenvalues, the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.