Conformal Geometry and Regularization of Disclinations by a Cosmological Constant in $(2+1)$ Dimensions
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| Format: | Preprint |
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2025
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| _version_ | 1866915474470600704 |
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| author | Carvalho, A. Furtado, C. |
| author_facet | Carvalho, A. Furtado, C. |
| contents | We investigate the effect of a cosmological constant $Λ$ on the geometry generated by a two-dimensional disclination in a conformal metric framework. For $Λ>0$, we obtain an exact analytic solution of the Liouville-type equation, which regularizes the defect core, preserves the topological charge, and yields a compact space with finite volume and positive curvature. For $Λ<0$, the solution must be obtained numerically and asymptotically approaches $R \to 3Λ< 0$, producing an open hyperbolic geometry with divergent volume.
In both regimes, the curvature profile is governed solely by the disclination strength $α$, while the sign of $Λ$ dictates the global phase: compact and confined for $Λ>0$, hyperbolic and delocalized for $Λ<0$. This establishes a clear geometric dichotomy and shows that the cosmological constant provides a natural analytic regularization beyond cutoff-based treatments, with implications for analog gravity and two-dimensional condensed matter systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal Geometry and Regularization of Disclinations by a Cosmological Constant in $(2+1)$ Dimensions Carvalho, A. Furtado, C. General Relativity and Quantum Cosmology We investigate the effect of a cosmological constant $Λ$ on the geometry generated by a two-dimensional disclination in a conformal metric framework. For $Λ>0$, we obtain an exact analytic solution of the Liouville-type equation, which regularizes the defect core, preserves the topological charge, and yields a compact space with finite volume and positive curvature. For $Λ<0$, the solution must be obtained numerically and asymptotically approaches $R \to 3Λ< 0$, producing an open hyperbolic geometry with divergent volume. In both regimes, the curvature profile is governed solely by the disclination strength $α$, while the sign of $Λ$ dictates the global phase: compact and confined for $Λ>0$, hyperbolic and delocalized for $Λ<0$. This establishes a clear geometric dichotomy and shows that the cosmological constant provides a natural analytic regularization beyond cutoff-based treatments, with implications for analog gravity and two-dimensional condensed matter systems. |
| title | Conformal Geometry and Regularization of Disclinations by a Cosmological Constant in $(2+1)$ Dimensions |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2509.01635 |