New conformal-metric solutions for continuous distributions of disclination-like defects

We develop a conformal formulation for two-dimensional geometries sourced by continuous distributions of topological defects, reducing the problem to a Poisson equation for the conformal factor. Throughout, we focus on radially symmetric, curvature-type (disclination-like) defect distributions in a torsion-free geometry. Closed-form solutions are obtained for Gaussian, exponential, and powerlaw (1/r) profiles. The Gaussian profile yields a regular core and an asymptotically flat far field. The exponential profile is treated on the punctured plane (r > 0), producing sharply localized curvature with finite total flux. For the 1/r case, we work on the punctured plane (or on annuli), obtaining an exterior logarithmic solution that encodes a conical geometry; more general power-law profiles require finite domains or decay to ensure integrability. In all cases, Gauss–Bonnet on annuli provides a global relation equating total curvature (holonomy/deficit angle) to the integrated defect density, while the δ-limit reproduces the standard conical geometry with distributional curvature. The framework offers a unified, physically motivated description of finite-core disclination-like defects relevant to analog gravity and two-dimensional material systems.