We found several new Oka manifolds. Among the new examples are Markov cubic surfaces in C3. Their algebraic automorphism groups are known to be discrete and act transitively on the set of points with integer components. By methods of Oka theory we showed that the tangent bundles of these surfaces are generated by algebraic vector fields, so their holomorphic automorphism groups are non-discrete and act transitively.
We found surprisingly small pseudoconcave Oka domains in Euclidean spaces Cn, n > 1. Indeed, most concave domains in Cn which are only slightly bigger than a halfspace are Oka. If E is a closed set in Cn whose projective closure avoids a hyperplane L and is polynomially convex in the complement of L, then the complement of E in Cn is an Oka domain. If E is a closed convex set in Cn contained in a closed halfspace H such that E ∩ bH is nonempty and bounded then Cn \E contains images of proper holomorphic maps X → Cn from any Stein manifold X of dimension < n. We constructed proper holomorphic embeddings of any Stein manifold of dimension n ≥ 1 in a complex Euclidean space of dimension 2n+1 with surprisingly small limit sets.
We studied Oka properties of disc tubes in semipositive hermitian line bundles (E,h) on a compact complex manifold X of dimension >1. Assuming that each point x ∈ X admits a divisor D whose complement X \ D is a Stein neighbourhood of x with the density property, we proved that the disc bundle {e ∈ E : |e| < 1} is Oka. This holds for any ample line bundle on a rational homogeneous manifold of dimension > 1, a class of manifolds including all projective spaces, Grassmannians, and flag manifolds. This contributes to the heuristic principles that the Oka property is related to metric positivity and that Oka domains are pseudoconcave.
Complex curves are of special importance in complex geometry. We introduced a new class of Oka-1 manifolds satisfying the Oka properties for maps from all open Riemann surfaces. We showed that a complex manifold densely dominable by a Euclidean space is an Oka-1 manifold. In particular, all Kummer surfaces and all elliptic K3 surfaces are Oka-1 manifolds. The class of Oka-1 manifolds is invariant under Oka maps inducing a surjective homomorphism of fundamental groups. We studied the algebraic version of the Oka-1 condition, showing that it is a birational invariant for compact algebraic manifolds and holds for all rational manifolds.
We developed the Oka theory for maps from families of complex structures on smooth open surfaces to any Oka manifold. We expect that similar results can be obtained for suitably tame families of Stein manifolds of higher dimension. These new techniques connect Oka theory to Teichmüller theory. They enable the construction of continuous or smooth families of directed holomorphic immersions, minimal surfaces, holomorphic Legendrian curves, etc.
We obtained the homotopy principle for group equivariant minimal surfaces in Euclidean spaces Rn for discrete groups acting on an open Riemann surface by holomorphic automorphisms, and acting on Rn by rigid maps. This seems to be the first results of its kind for surfaces of higher genus. The main problem was to control periods of their derivative maps in connection with the group equivariance property.
A new Kobayashi-type intrinsic metric on domains in Euclidean spaces was introduced by using conformal minimal discs. It describes the fastest rate of growth of minimal surfaces in the domain. We obtained new results on Gromov hyperbolicity for the Kobayashi and minimal metrics. It was shown that every bounded strongly minimally convex domain in Rn is Gromov hyperbolic.
