Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

HOLOMORPHIC AUTOMORPHISMS AND PROPER HOLOMORPHIC SELF-MAPPINGS OF A TYPE OF GENERALISED MINIMAL BALL

In this paper, we first give a description of the holomorphic automorphism group of a convex domain which is a simple case of the so-called generalised minimal ball. As an application, we show that any proper holomorphic self-mapping on this type of domain is biholomorphic.

K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS

We show that any $n$-dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

On the holomorphic automorphism group of the Bergman–Hartogs domain

The Bergman–Hartogs domain which can be regarded as a generalization of the Cartan–Hartogs domain provides a large class of bounded pseudoconvex domains which are in general nonhomogeneous. Since the geometry of a domain is determined by its automorphism group to a certain extent, it is meaningful to study the structure of the automorphism group. In this paper, we completely determine the structure of the holomorphic automorphism group of the Bergman–Hartogs domain over a minimal homogeneous domain with center at the origin.