Families of proper holomorphic maps.

Area of Science:

• Complex Geometry
• Differential Geometry
• Topology

Background:

• Smooth, open, oriented surfaces (X) are fundamental objects in geometry.
• Complex structures define how to locally view a surface as a complex manifold.
• A continuous family of complex structures allows for smooth transitions between different geometric viewpoints.

Purpose of the Study:

• To construct a continuous family of proper holomorphic maps.
• To connect a surface with varying complex structures to a fixed complex space (C^2).
• To explore the relationship between families of complex structures and geometric mappings.

Main Methods:

• Utilizing the properties of smooth, open, oriented surfaces.
• Employing the concept of a continuous family of complex structures parameterized by a metrisable space B.
• Constructing proper holomorphic maps (F_b) that depend continuously on the parameter b.

Main Results:

• A continuous family of proper holomorphic maps (F_b) was successfully constructed.
• These maps link the surface (X) with its continuously varying complex structures (J_b) to the complex space C^2.
• The construction demonstrates the existence of such maps for each parameter b in B.

Conclusions:

• The study establishes a method for creating continuous families of proper holomorphic maps.
• This work provides a bridge between families of complex structures on surfaces and mappings into complex Euclidean space.
• The findings have implications for understanding the deformation of complex structures and related geometric properties.