Conformal Geometry and Regularization of Disclinations by a Cosmological Constant in (2+1) Dimensions

We investigate the effect of a cosmological constant Λ on the geometry generated by a two-dimensional disclination within a conformal metric framework. For Λ > 0, we obtain an exact analytic solution of the Liouville-type equation, providing a natural regularization of the defect while preserving its local conical structure. The resulting geometry possesses constant positive scalar curvature, R = 3Λ. For Λ < 0, no real closed-form solution was found within the present approach; the corresponding numerical solutions approach an asymptotically hyperbolic geometry with R → 3Λ < 0. The analysis shows that the cosmological constant determines the curvature scale and asymptotic behavior of the regularized geometry, while the disclination strength α remains encoded in its local topology. These results establish a clear geometric distinction between the Λ > 0 and Λ < 0 regimes and demonstrate that the cosmological constant provides a natural regularization mechanism beyond cutoff-based approaches, with potential applications to analog gravity and geometric models of condensed-matter systems.