New Solutions for Topological Defects with Continuous Distributions:A Conformal Metric Perspective

TL;DR

This paper develops exact solutions for two-dimensional geometries with continuous topological defect distributions using a conformal metric approach, revealing how smooth defect profiles regularize curvature singularities and encode topological features.

Contribution

It introduces a unified method to model regularized topological defects with continuous distributions in 2D geometries via a conformal metric framework, solving Einstein's equations as a Poisson problem.

Findings

Smooth defect profiles regularize curvature singularities.

Geometries interpolate between defect cores and flat regions.

Results align with the Gauss-Bonnet theorem.

Abstract

We present new exact solutions for two-dimensional geometries generated by continuous distributions of topological defects within a conformal metric framework. By reformulating Einstein's equations in two dimensions as a Poisson equation for the conformal factor, we analyze how smooth defect densities -- such as Gaussian, exponential, and power-law profiles -- regularize curvature singularities and encode nontrivial topological information. Each distribution yields a well-defined geometry that interpolates between localized curvature near the defect core and asymptotic flatness. We compute the Ricci scalar and total curvature, confirming consistency with the Gauss-Bonnet theorem. Our results provide a unified geometric description of regularized disclination-like defects and offer insights into analog gravity, crystalline materials, and two-dimensional systems with emergent curvature.