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Register a new userStably-interior points and the Semicontinuity of the Automorphism group
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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.
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We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.
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.
We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<lfloorfrac{n}{2}rfloor$, then it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdans property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).
In this note, we consider the sufficient coefficient condition for some harmonic mappings in the unit disk which can be extended to the whole complex plane. As an application of this result, we will prove that a harmonic strongly starlike mapping has a quasiconformal extension to the whole plane and will give an explicit form of its extension function. We also investigate the quasiconformal extension of harmonic mappings in the exterior unit disk.
The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.
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