No abstract uploaded!

No abstract uploaded!

Let$D$be a convex domain with smooth boundary in complex space and let$f$be a continuous function on the boundary of$D$. Suppose that$f$holomorphically extends to the extremal discs tangent to a convex subdomain of$D$. We prove that f holomorphically extends to$D$. The result partially answers a conjecture by Globevnik and Stout of 1991.

We construct a 2-dimensional complex manifold$X$which is the increasing union of proper subdomains that are biholomorphic to ℂ^{2}, but$X$is not Stein.

Let g be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra U(g) has intermediate growth and thus infinite Gelfand–Kirillov (GK-) dimension. We prove that the GK-dimension of U(g) is just infinite in the sense that any proper quotient of U(g) has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra S(g) by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.