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Register a new userA characterization of the bidisc by a subgroup of its automorphism group
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We make a connection between the structure of the bidisc and a distinguished subgroup of its automorphism group. The automorphism group of the bidisc, as we know, is of dimension six and acts transitively. We observe that it contains a subgroup that is isomorphic to the automorphism group of the open unit disc and this subgroup partitions the bidisc into a complex curve and a family of strongly pseudo-convex hypersurfaces that are non-spherical as CR-manifolds. Our work reverses this process and shows that any $2$-dimensional Kobayashi-hyperbolic manifold whose automorphism group (which is known, from the general theory, to be a Lie group) has a $3$-dimensional subgroup that is non-solvable (as a Lie group) and that acts on the manifold to produce a collection of orbits possessing essentially the characteristics of the concretely known collection of orbits mentioned above, is biholomorphic to the bidisc. The distinguished subgroup is interesting in its own right. It turns out that if we consider any subdomain of the bidisc that is a union of a proper sub-collection of the collection of orbits mentioned above, then the automorphism group of this subdomain can be expressed very simply in terms of this distinguished subgroup.
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In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.
The first result is the semicontinuity of automorphism groups for the collection of complex two-dimensional bounded pseudoconvex domains with smooth boundary of finite DAngelo type. The method of proof is new so that it simplifies the previous proof of earlier semicontinuity theorems on bounded strongly pseudoconvex daomains by Greene and Krantz in the early 1980s.
We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaevs classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain [{(z_1,z_2)in mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+overline{z_2}))>0}] in Isaevs list. Isaev calls it $mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $mathcal D_1$. Among the results on $mathcal D_1$, of particular interest is the fact that $mathcal D_1$ is a symmetrization. When we symmetrize (appropriately defined in the context in the last section) either $Omega_1$ or $mathcal{D}^{(2)}_1$ (Isaevs notation), we get $mathcal D_1$. These two domains $Omega_1$ and $mathcal{D}^{(2)}_1$ are in Isaevs list and he mentioned that these are biholomorphic to $mathbb{D} times mathbb{D}$. We produce explicit biholomorphisms between these domains and $mathbb{D} times mathbb{D}$.
We will show a rigidity of a Kahler potential of the Poincare metric with a constant length differential.
Any group $G$ gives rise to a 2-group of inner automorphisms, $mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} to mathrm{INN}_0(G_{(2)}) to Sigma G_{(2)}$. As a consequence, $mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.
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