Undulating Conformal Boundaries in 3D Gravity

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 $\Lambda$, 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 $\Lambda < 0$ and $2<K|\Lambda|^{-1/2}<3/\sqrt{2}$. Moreover, for all values of $\Lambda$, there exist patches of space with a non-contractible thermal circle and a macroscopic entropy. We further analyse the problem in both the AdS$_3$ boundary limit and the stretched dS$_3$ horizon limit, and comment on a recasting of the problem in terms of classical strings.