  Recently, remarkable progress in recovering the Page curve of an evaporating
black hole (BH) in Jackiw-Teitelboim gravity has been achieved through use of
Quantum Extremal surfaces (QES). Multi-boundary Wormhole (MbW) models have been
crucial in parallel model building in three dimensions. Motivated by this we
here use the latter models to compute the subregion complexity of the Hawking
quanta of the evaporating BH in AdS$_{3}$ and obtain the Page curve associated
with this information theoretic measure. We use three- and $n$-boundary
wormhole constructions to elucidate our computations of volumes below the
Hubeny-Rangamani-Takayanagi (HRT) surfaces at different times. Time is
represented by the growing length of the throat horizons corresponding to
smaller exits of the multi-boundary wormhole and the evaporating bigger exit
shrinks with evolving time. We track the change in choice of HRT surfaces with
time and plot the volume with time. The smooth transition of Page curve is
realized by a discontinuous jump at Page time in volume subregion complexity
plots and the usual Page transition is realized as a phase transition due to
the inclusion of the island in this context. We discuss mathematical
intricacies and physical insights regarding the inclusion of the extra volume
at Page time. The analysis is backed by calculations and lessons from kinematic
space and tensor networks.
