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In this paper we study the rigidity of proper holomorphic maps $f\colon \Omega\to\Omega'$ between irreducible bounded symmetric domains $\Omega$ and $\Omega'$ with small rank differences: $2\leq \text{rank}(\Omega')< 2\,\text{rank}(\Omega)-1$. More precisely, if either $\Omega$ and $\Omega'$ have the same type or $\Omega$ is of type~III and $\Omega'$ is of type~I, then up to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\times F_2\colon \Omega\to \Omega_1'\times \Omega_2'$. Here $\Omega_1'$, $\Omega_2'$ are bounded symmetric domains, the map $F_1\colon \Omega \to \Omega_1'$ is a standard embedding, $F_2: \Omega \to \Omega_2'$, and $\imath\colon \Omega'_1\times \Omega'_2 \to \Omega'$ is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map $f: \Omega \to \Omega'$ if $\Omega$ is of type~I and $\Omega'$ is of type~III, or $\Omega$ is of type~II and $\Omega'$ is either of type~I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of $\Omega$, we construct rational maps between moduli spaces of subgrassmannians of compact duals of $\Omega$ and $\Omega'$, and induced CR-maps between CR-hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.

Let $D_{p,q}$ and $D_{p',q'}$ be irreducible bounded symmetric domains of the first kind with rank $q$ and $q'$, respectively and let $f:D_{p,q}\to D_{p',q'}$ be a proper holomorphic map that extends $C^2$ up to the boundary. In this paper we show that if $q, q'\geq 2$ and $f$ maps Shilov boundary of $D_{p,q}$ to Shilov boundary of $D_{p',q'}$, then $f$ is of the form $f = \imath\circ F$, where $$ F=F_1\times F_2\colon D_{p,q}\to \Omega_1'\times \Omega_2', $$ $\Omega_1'$ and $\Omega_2'$ are bounded symmetric domains, $F_1 \colon D_{p,q}\to \Omega_1'$ is a proper rational map, $F_2:D_{p,q}\to \Omega_2'$ is not proper and $\imath: \Omega_1' \times \Omega'_2 \hookrightarrow D_{p',q'}$ is a holomorphic totally geodesic isometric embedding of a reducible bounded symmetric domain $\Omega_1' \times \Omega_2'$ into $D_{p',q'}$ with respect to canonical K\"ahler-Einstein metrics. Moreover, if $p>q$, then $f$ is a rational map. As an application we show that a proper holomorphic map $f:D_{p,q}\to D_{p',q'}$ that extends $C^\infty$ up to the boundary is a rational map or a totally geodesic isometric embedding with respect to the Kobayashi metrics, if $3\leq q \leq q'\leq 2q-1.$

In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon \Omega\to\Omega'$ where $\Omega$, $\Omega'$ are bounded symmetric domains, if $\Omega$ is irreducible and $\text{rank}(\Omega) \geq \text{rank}(\Omega')$ or more generally, $\text{rank}(\Omega) \geq \text{rank}(f_*v)$ for any tangent vector $v$ of $\Omega$, then $f$ is either holomorphic or anti-holomorphic. Secondly we characterize $C^1$ Kobayashi isometries from a reducible bounded symmetric domain to itself., Comment: 20 pages

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon \Omega\to\Omega'$ between bounded symmetric domains $\Omega$ and $\Omega'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $\Omega$ is irreducible, there exist totally geodesic bounded symmetric subdomains $\Omega_1$ and $\Omega_2$ of $\Omega'$ such that $f = (f_1, f_2)$ maps into $\Omega_1\times \Omega_2\subset \Omega$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}(\Omega')<2\text{rank}(\Omega)$, then either $f$ or $\bar f$ is a standard holomorphic embedding., Comment: 10 pages

We provide a solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers for $(0,q)$ forms for arbitrary $q$. As an application, we obtain subelliptic estimates for $(0,q)$ forms with effectively controlled order $\epsilon>0$ (the Sobolev exponent) for domains given by sums of squares of holomorphic functions (J.J. Kohn called them "special domains"). These domains are of particular interest due to their relation with complex and algebraic geometry. Our methods include triangular resolutions introduced by the authors in their previous work., Comment: arXiv admin note: substantial text overlap with arXiv:2003.06482

A solution to the effectiveness problem in Kohn's algorithm for generating subelliptic multipliers is provided for domains that include those given by sums of squares of holomorphic functions (also including infinite sums). These domains are of particular interest due to their relation with complex and algebraic geometry and in particular, seem to include all previously known cases. Furthermore, combined with a recent result of M. Fassina, our effectiveness method allows establishing effective subelliptic estimates for more general classes of domains. Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as $Q\circ\Gamma$, where $\Gamma$ is constructed from pre-multipliers and $Q$ is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic-geometric techniques by Y.-T. Siu and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo. The proposed procedures are purely algebraic and as such can also be of interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

Orbits of $SU(\ell, m)$ in a Grassmannian manifold have homogeneous CR structures. In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian into another Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a smooth CR mapping sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ in Grassmannian manifolds to extend as a totally geodesic embedding of the Grassmannian into another Grassmannian.

We propose a new class of geometric invariants called jet vanishing orders, and use them to establish a new selection algorithm in the Kohn's construction of subelliptic multipliers for special domains in dimension $3$, inspired by the work of Y.-T. Siu [S10]. In particular, we obtain effective termination of our selection algorithm with explicit bounds both for the steps of the algorithm and the order of subellipticity in the corresponding subelliptic estimates. Our procedure possesses additional features of certain stability under high order perturbations, due to deferring the step of taking radicals to the very end of the algorithm. We further illustrate by examples the sharpness in our technical results and demonstrate the complete procedure for arbitrary high order perturbations of the Catlin-D'Angelo example [CD10] in Section 5. Our techniques here may be of broader interest for more general PDE systems, in the light of the recent program initiated by the breakthrough paper of Y.-T. Siu [S17]., Comment: Final version as published, substantially revised introduction and expanded literature, examples added in the last section to illustrate the algorithm

Our first main result gives assumptions guaranteeing that proper holomorphic maps between Cartan type I bounded symmetric domains have simple block matrix shape, answering positively a question of Mok. The proof is based on the second main result establishing similar phenomenon for local CR maps between arbitrary boundary components of two bounded symmetric domains of the above type. Since boundary components other than Shilov boundaries are Levi-degenerate, our analysis is based on their $2$-nondegeneracy combining Levi forms with higher order tensors.

Our goal is to establish what seems to be the first rigidity result for CR embeddings between Shilov boundaries of bounded symmetric domains of higher rank. The result states that any such CR embedding is the standard linear embedding up to CR automorphisms. Our basic assumption extends precisely the well-known optimal bound for the rank one case. There are no other restrictions on the ranks, in particular, the difficult case when the target rank is larger than the source rank is also allowed.

A method is proposed to obtain examples of smooth CR-manifolds whose local stability group is neither a Lie group nor infinite-dimensional., Comment: Final version, some proofs are simplified, examples added, to appear in Compositio Mathematica

We give a solution to the equivalence and the embedding problems for smooth CR-submanifolds of complex spaces (and, more generally, for abstract CR-manifolds) in terms of complete differential systems in jet bundles satisfied by all CR-equivalences or CR-embeddings respectively (local and global). For the equivalence problem, the manifolds are assumed to be of finite type and finitely nondegenerate. These are higher order generalizations of the corresponding nondegeneracy conditions for the Levi form. It is shown by a simple example that these nondegeneracy conditions cannot be even slightly relaxed to more general known conditions. In particular, for essentially finite hypersurfaces in $\C^2$, such a complete system may not exist in general. For the embedding problem, the source manifold is assumed to be of finite type and the embeddings to be finitely nondegenerate. Situations are given, where the last condition is automatically satisfied by all CR-embeddings.