Undulating Conformal Boundaries in 3D Gravity
Abstract
We consider three-dimensional Einstein gravity in Euclidean signature with a finite boundary of torus topology endowed with an induced metric of fixed conformal class and a constant trace of extrinsic curvature K. For vanishing, positive, and negative cosmological constant Λ, we analytically determine boundaries enclosing different patches of locally flat, de Sitter (dS3), and Anti-de Sitter (AdS3) spaces. We find solutions that depend non-trivially on either cycle of the torus, noting that some of them exhibit self-intersections. Adapting the Gibbons-Hawking prescription of interpreting the Euclidean gravitational path integral as a thermal partition function, we explore the rich semi-classical thermodynamic phase space of the problem. While most saddles are found to be either thermally unstable or metastable compared to those with uniform boundaries, we find inhomogeneous solutions that are thermodynamically favourable in the case of Λ< 0 and 2<K|Λ|−1/2<3/√2. Moreover, for all values of Λ, there exist patches of space with a non-contractible thermal circle and a macroscopic entropy. We comment on a recasting of the problem in terms of classical strings. We further analyse the problem in both the stretched dS3 horizon limit and the AdS3 boundary limit. Using the results for the latter, we find limitations to a proposed holographic dual that consists of a two-dimensional holographic conformal field theory coupled to timelike Liouville, deformed by a marginal TT̅-type operator.
Related
- Boundaries and defects in conformal field theories at finite temperature
- Conformal boundaries and the photon sphere of AdS black holes
- On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity
- On integrable structures on non–compact boundaries in three–dimensional gravity
- Conformal Renormalisation of 8D Einstein Gravity
BibTeX
@misc{hamdan2026undulating,
title = {{Undulating Conformal Boundaries in 3D Gravity}},
author = {Weam Abou Hamdan and Chawakorn Maneerat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.08058},
eprint = {2605.08058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08058},
}
RIS
TY - UNPB
TI - Undulating Conformal Boundaries in 3D Gravity
AU - Weam Abou Hamdan
AU - Chawakorn Maneerat
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2605.08058
UR - https://arxiv.org/abs/2605.08058
AB - We consider three-dimensional Einstein gravity in Euclidean signature with a finite boundary of torus topology endowed with an induced metric of fixed conformal class and a constant trace of extrinsic curvature K. For vanishing, positive, and negative cosmological constant Λ, we analytically determine boundaries enclosing different patches of locally flat, de Sitter (dS_3), and Anti-de Sitter (AdS_3) spaces. We find solutions that depend non-trivially on either cycle of the torus, noting that some of them exhibit self-intersections. Adapting the Gibbons-Hawking prescription of interpreting the Euclidean gravitational path integral as a thermal partition function, we explore the rich semi-classical thermodynamic phase space of the problem. While most saddles are found to be either thermally unstable or metastable compared to those with uniform boundaries, we find inhomogeneous solutions that are thermodynamically favourable in the case of Λ< 0 and 2<K|Λ|^−1/2<3/√2. Moreover, for all values of Λ, there exist patches of space with a non-contractible thermal circle and a macroscopic entropy. We comment on a recasting of the problem in terms of classical strings. We further analyse the problem in both the stretched dS_3 horizon limit and the AdS_3 boundary limit. Using the results for the latter, we find limitations to a proposed holographic dual that consists of a two-dimensional holographic conformal field theory coupled to timelike Liouville, deformed by a marginal TT̅-type operator.
ER -