Neck-pinching of -structures in the -character variety

Area of Science:

• Complex Geometry
• Topology
• Differential Geometry

Background:

• Characterizes neck-pinching degenerations of geometric structures on closed oriented surfaces of genus at least two.
• Considers paths of geometric structures whose holonomy converges in the character variety.
• Notes that the complex structure also leaves compact subsets of Teichmüller space.

Purpose of the Study:

• To describe the limit of geometric structures pinched along a loop on the surface.
• To offer different perspectives on the limit, including developing maps, holomorphic quadratic differentials, and pleated surfaces.
• To investigate the nature of holonomy representations for these structures.

Main Methods:

• Analysis of geometric structures on surfaces with genus >= 2.
• Study of paths of structures with converging holonomy.
• Investigation under the condition of pinching along a loop.

Main Results:

• Provides a description of the limit of pinched geometric structures.
• Explores the limit through developing maps, holomorphic quadratic differentials, and pleated surfaces.
• Confirms that holonomy representations are nonelementary (strongly irreducible and unbounded).

Conclusions:

• The study offers a comprehensive characterization of neck-pinching degenerations.
• Presents an exotic example of a path whose limit holonomy is the trivial representation.
• Enhances understanding of geometric structures and their limiting behaviors on surfaces.