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The polydisk theorem for Hartogs domains over symmetric domains

Andrea Loi, Roberto Mossa, Fabio Zuddas

Abstract

We extend the polydisk theorem of [21], originally established for classical Cartan-Hartogs domains, to Hartogs domains over arbitrary (possibly reducible and exceptional) bounded symmetric domains. We further establish a dual counterpart of this result. As an application, we show that the dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold, thereby broadening the rigidity phenomena obtained in [13].

The polydisk theorem for Hartogs domains over symmetric domains

Abstract

We extend the polydisk theorem of [21], originally established for classical Cartan-Hartogs domains, to Hartogs domains over arbitrary (possibly reducible and exceptional) bounded symmetric domains. We further establish a dual counterpart of this result. As an application, we show that the dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold, thereby broadening the rigidity phenomena obtained in [13].

Paper Structure

This paper contains 10 sections, 12 theorems, 113 equations, 1 table.

Table of Contents

  1. Introduction
  2. Hermitian Symmetric Spaces and Polydisk Embeddings
  3. Cartan domains
  4. Hermitian positive Jordan triple system of type VI
  5. The exceptional Cartan domain of type VI
  6. The exceptional domain of type V
  7. Proofs of the main results
  8. Proof of Theorem \ref{['THMhpoly']}
  9. Proof of Theorem \ref{['THMhpolydual']}
  10. Proof of Theorem \ref{['THMCORhpolydual']}

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{C}^n$ be a bounded symmetric domain of rank $r$. Then for every point $p\in M_{\Omega,\mu}$ and every vector $X\in T_{p}M_{\Omega,\mu}$, there exist totally geodesic holomorphic embeddings such that $\widetilde{j}$ is an extension of $j$, $\widetilde{j}(q)=p$ and $(d\widetilde{j})_{q}(V)=X$ for some $q\in M_{\Delta^{r},\mu}$ and $V\in T_qM_{\Delta^{r},\mu}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 12 more