Discrete Yamabe Problem for Polyhedral Surfaces.

Area of Science:

• Differential Geometry
• Computational Geometry
• Discrete Differential Geometry

Background:

• Polyhedral surfaces lack a standard definition for Gaussian curvature at singularities.
• Existing methods for discrete conformal equivalence are limited.

Purpose of the Study:

• To define a novel discrete Gaussian curvature for polyhedral surfaces.
• To explore discrete conformal classes and the existence of surfaces with constant discrete Gaussian curvature.

Main Methods:

• Defining discrete Gaussian curvature as the ratio of angle defect to Voronoi cell area at conical singularities.
• Generalizing discrete conformal equivalence to partition surfaces into discrete conformal classes.

Main Results:

• A new definition for discrete Gaussian curvature on polyhedral surfaces is established.
• The existence of a polyhedral surface with constant discrete Gaussian curvature within every discrete conformal class is proven.
• Examples demonstrate that such surfaces are not always unique.

Conclusions:

• The developed discrete Gaussian curvature provides a new tool for analyzing polyhedral surfaces.
• The concept of discrete conformal classes is extended, revealing structural properties of these surfaces.
• The existence and non-uniqueness of constant curvature surfaces offer avenues for further geometric research.