J1-3005 Complex and geometric analysis

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Research project is (co) funded by the Slovenian Research Agency.

UL Member: Faculty of Mathematics and Physics

Code: J1-3005

Project: Complex and geometric analysis

Period: 1. 10. 2021 - 30. 9. 2024

Range per year: 0,8 FTE, category: B

Head: Franc Forstnerič

Research activity: Natural sciences and mathematics

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Project description:

We propose problems in geometric complex analysis, complex dynamics, and numerical approximation theory in the following areas:

(A) Oka manifolds and their role in complex geometry

(B) Cauchy‑Riemann singularities of real manifolds in complex manifolds

(C) Riemann‑Hilbert boundary value problems

(D) Gluing techniques on almost complex manifolds

(E) Problems in complex dynamics

(F) Approximation of curves and surfaces by polynomial objects

(A) Oka manifolds were introduced in the literature by the PI in 2009. This class of complex manifolds has since become the focus of holomorphic flexibility theory and was assigned the new MSC‑2020 field 32Q56 Oka principle and Oka manifolds. An Oka manifold admits many holomorphic images from any Stein manifold, which makes them highly interesting objects dual to Stein manifolds. Complex manifolds which are sufficiently symmetric, including many model manifolds, are Oka. In a string of developments since 2018, Kusakabe gave a new characterization of Oka manifolds which shows that they are much more prevalent than previously thought. We shall explore these new directions, combining the recently discovered results and techniques with those from the PI’s 2017 monograph Stein Manifolds and Holomorphic Mappings. We will look for a geometric characterization of Oka domains in Cn, explore the position of Oka manifolds among compact projective manifolds and their relationship with Campana special manifolds and with metric positivity. We will also look for large Euclidean domains in Stein manifolds and explore possible degenerations of Euclidean fibres in Stein fibrations.

(B) A major aspect of Cauchy–Riemann (CR) geometry concerns understanding the structure of CR singular points of smooth n‑manifolds in complex n‑manifolds. The situation is well understood for real surfaces in complex surfaces (n=2) and to some extent in n=3, but not for n>3. We shall investigate which 2‑surfaces or their links arise as CR singular points of immersions or embeddings of 4‑spheres into C4. We shall also consider the structure and classification of complex points in real codimension 2 submanifolds of complex manifolds, focusing on the stability and classification of the quadratic part of such points. The notions of CR nonminimal and CR nondegenerate points play an important role in this analysis.

(C+D) The Riemann‑Hilbert boundary value problem provides a useful method for solving geometric problems of complex analysis and related fields. We shall develop this technique for applications to complex contact manifold and, via the Penrose twistor space theory, to superminimal surfaces in self‑dual Einstein 4‑manifolds. We will also develop gluing methods for holomorphic curves in almost complex manifolds of dimension >2, offering a strong tool for the construction of proper curves and solution of the Calabi‑Yau problem on such manifolds.

(E) We will study dynamics of holomorphic maps of several variables with the aim of solving the following problems: (1) Obtain a complete classification of invariant Fatou components for holomorphic automorphisms of C2 for some of their subclasses such as Hénon maps, (2) classify the possible dynamical behavior inside a wandering domain for holomorphic automorphisms of C2, (3) obtain a complete description of a local dynamics of skew‑products (z,w) → (p(z),q(z,w)) tangent to the identity.

(F) Smooth curves and surfaces are important mathematical objects in research and engineering. They can be used to describe many structures and are particularly important in computer aided geometric design and computer aided manufacturing. Curves and surfaces must often be represented in a polynomial form in order to enable efficient visualization and computing. Since even most basic objects such as circular arcs or spheres do not have polynomial representation, approximation by polynomial objects is unavoidable. We propose to develop effective new methods for polynomial approximation of curves and surfaces.https://www.sicris.si/public/jqm/search_basic.aspx?lang=eng&opt=2&subopt=403&opdescr=search&code1=cmn&code2=auto&search_term=J1-3005