Extremal surfaces in glue-on AdS/$T\bar T$ holography

$T\bar T$ deformed CFTs with positive deformation parameter have been proposed to be holographically dual to Einstein gravity in a glue-on $\mathrm{AdS}_3$ spacetime. The latter is constructed from AdS$_3$ by gluing a patch of an auxiliary AdS$_3^*$ spacetime to its asymptotic boundary. In this work, we propose a glue-on version of the Ryu-Takayanagi formula, which is given by the signed area of an extremal surface. The extremal surface is anchored at the endpoints of an interval on a cutoff surface in the glue-on geometry. It consists of an RT surface lying in the AdS$_3$ part of the spacetime and its extension to the AdS$_3^*$ region. The signed area is the length of the RT surface minus the length of the segments in AdS$_3^*$. We find that the Ryu-Takayanagi formula with the signed area reproduces the entanglement entropy of a half interval for $T\bar T$-deformed CFTs on the sphere. We then study the properties of extremal surfaces on various glue-on geometries, including Poincar\'e $\mathrm{AdS}_3$, global $\mathrm{AdS}_3$, and the BTZ black hole. When anchored on multiple intervals at the boundary, the signed area of the minimal surfaces undergoes phase transitions with novel properties. In all of these examples, we find that the glue-on extremal surfaces exhibit a minimum length related to the deformation parameter of $T\bar T$-deformed CFTs.