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Research group in Complex Analysis and Analytical Geometry

Complex analysis and analytical geometry, geometry of the CR varieties, of the Levi-flat varieties and related non-linear problems. Holomorphic envelopes and Levi problems. Structure of weakly pseudo-complete complex spaces.

Research Areas and Themes

A) Existence of holomorphic chains and Levi flat hypersurfaces with assigned boundary, in the context of complex structures and, more generally, in the context of quasi complex structures .

Levi flat hypersurfaces with assigned boundary come out in the study of the holomorphic envelope of real subvarieties in a complex space. Research about this has primarily been carried out in the case in which the ambient space is a two dimension Stein manifold using the analytic discs method of Bishop or the Gromov's compactness theorem for J-holomorphic curves.

The geometric problem can be translated in a Dirichlet problem for a quasi-linear, degenerate elliptic equation, the Levi equation for the quasi complex structure.

The problem has been studied recently in the context of certain non calibrated, quasi complex structures on R4. The existence theorem is obtained by using fairly sophisticated regularization techniques for the weak viscose solution of the Levi operator.

When the ambient space is a complex manifold of higher dimension, different from the case above, the problem is overdetermined.

As yet unpublished results have been obtained recently for the space Cn.

B) Existence of Levi-flat hypersurfaces with a partly-assigned boundary
Research on this topic has begun very recently, and partial results have been proved for the space C2. They are particularly interesting and can be considered a first step towards a “theory of domains of existence” for flat Levi hypersurfaces.

C) Geometric Structure of the weakly complete complex spaces.
The study of the geometric properties of a “weakly complete” complex X space, i.e., one which allows a weakly plurisubharmonic exhaustion function, can be quite difficult even if X is of dimension 2. It presents a number of problems of the existence on X of global analytical objects (holomorphic functions, complex subspaces, Levi-flat hypersurfaces, etc.). Preliminary results have been recently obtained with the hypothesis in which (X is of dimension 2 and) the plurisubharmonic exhaustion function is real-analytic. It is a strong hypothesis and sufficient conditions to guarantee its existence are unknown.

D) Existence of the envelope of holomorphy for open complex spaces and the Levi problem.
It is a classic from the theory of functions of several complex variables. The last significant results date back to the beginning of the ‘60s.

E) Evolution of compact subsets of C2 and CP2 by Levi form
Among the motivations for research on this theme there is on one hand the possibility of obtaining information on the various envelopes of a compact K and on the other the analysis of the singularities which can be obtained for evolution, beginning with regular sets. The problem is closely connected to A) above.
It has been shown that the evolution by Levi form of a graph ∑ in C2, over a bounded domain D in C x R, has bounded the Levi flat hypersurface with boundary b as a limit.