Complex and Riemannian Geometry and their Applications

Project number: KP-06-Н82/6

Base organization:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Project leader:
Prof. PhD Velichka Milousheva

Funding:
Bulgarian National Science Fund, Competition for financial support of basic research projects – 2024

Period:
December 2024 – December 2027

Team:

Prof. PhD Velichka Milousheva

IMI-BAS

Corr. Member of BAS Prof. D.Sc. Nikolai Nikolov

IMI-BAS

Assist. Prof. PhD Maria Trybula

Adam Mickiewicz University in Poznan and IMI-BAS

Chief Assist. Prof. PhD Victoria Bencheva

IMI-BAS and VTU “St. St. Cyril and Methodius”

Prof. D.Sc. Johann Davidov

IMI-BAS

Prof. PhD Ognian Kassabov

IMI-BAS

Research plan:

Work package 1.

Analytic and geometric problems in several complex variables

Work package 2.

Geometry of generalized twistor spaces

Work package 3.

The problem of Lund-Regge for surfaces in 4-dimensional pseudo-Euclidean spaces

Expected results:

Finding an optimal regularity of the boundary of a domain in Cⁿ for which Kobayashi visibility implies pseudoconvexity.
A representation of bounded linear operators between spaces of holomorphic functions for certain classes of domains in Cⁿ via germs of holomorphic functions.
Establishing the possible Gray-Hervella classes of natural almost Hermitian structures on a generalized twistor space.
Obtaining a coordinate-free formula for the curvature of a natural Riemannian metric on а generalized twistor space and finding conditions on the base manifold for this metric to be Einstein.
Finding conditions on two generalized Riemannian metrics on a manifold under which the generalized twistor spaces determined by these metrics “coincide “, i.e. are equivalent in a suitable sense, when considering with their generalized almost complex structures.
Describing the class of marginally trapped (quasi-minimal) surfaces in terms of minimal number of partial differential equations.
Introducing special uniquely determined parameters on surfaces in the Euclidean 4-space that allow the reduction of the number of PDEs determining the surfaces.
Introducing special isotropic parameters on Lorentz surfaces with parallel normalized mean curvature vector field in the pseudo-Euclidean 4-space with neutral metric and solving the Lund-Regge problem.
Construction of meridian timelike surfaces in the Minkowski 4-space and investigation of their basic invariants.

Results obtained:

It is proved that the weak triangle inequality for the Lempert function of a domain in в $\mathbb C^n$ with $C^1$ -smooth boundary is a necessary condition for the pseudoconvexity of the domain.
The exact asymptotics of the Kobayashi distance around strictly pseudoconvex boundary points and around non-pseudoconvex boundary points of bounded domains in $\mathbb C^n$ with $C^2$ -smooth boundaries are found.
Exact estimates for invariant metrics of model non-pseudoconvex domains in $\mathbb C^2$ are obtained.
A characterization of Hadamard multipliers on the space H(Ω), where Ω ⊂ $\mathbb C^n$ is a Runge domain, is obtained by means of analytic functionals.
A theorem for representing bounded linear operators between spaces of holomorphic functions on cylindrical domains is proved.
Global formulas for the covariant derivatives of two horizontal vector fields, two vertical fields, and a pair consisting of a horizontal and a vertical field are derived; these formulas are used to calculate the covariant derivatives of the almost Hermitian structures defined on a generalized twistor space, as well as the corresponding Nijenhuis tensors.
The possible Gray-Hervela classes for these Hermitian structures are established.
Global formulas for components of the curvature tensor on a generalized twistor space are obtained.
It is shown that in the general case, any surface in the 4-dimensional Euclidean space admits (at least locally) canonical parameters, which allow reducing the number of invariant functions determining the surface up to a motion. This solves the Lund-Regge problem for the general class of surfaces in the 4-dimensional Euclidean space.
It is proved that every marginally trapped surface of a general type in the 4-dimensional Minkowski space locally admits canonical parameters that allow solving the Lund-Regge problem for this class of surfaces.
The following main classes of timelike meridian surfaces of elliptic or hyperbolic type are described: with constant Gaussian curvature, with constant mean curvature, with parallel mean curvature vector field, with parallel normalized mean curvature vector field.
A classification of some important geometric objects on general rotational surfaces in the pseudo-Euclidean 4-dimensional space with neutral metric are obtained, such as Killing vector fields, homothetic vector fields, divergence-free vector fields, co-closed and harmonic one-forms, and also harmonic functions.