Holomorphic isometries from the Poincar\'e disk into bounded symmetric domains of rank at least two
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.
In this paper, we establish the local well-posedness for a new coupled Camassa-Holm system in a range of the Besov spaces by employing a series of norm estimates. A wave-breaking mechanism for solutions is described in detail and a result of blow-up solutions with certain profile is established.