  We study false vacuum decay in a black hole (BH) spacetime with an angular
momentum. Considering the false vacuum region described by a Kerr-de Sitter
geometry, under the thin wall approximation, we can obtain the stationary
configuration of the vacuum bubble seen from the outside false vacuum region
without specifying the geometry inside the domain wall. Then, assuming the true
vacuum region is described by a Kerr geometry, we can fix the mass and the spin
parameter for the Kerr geometry by imposing the 1st junction conditions and
conservation of the angular momentum. Although the assumption of the Kerr
geometry inside the domain wall cannot be fully consistent with the 2nd
junction conditions, we can roughly evaluate the error associated with this
inconsistency by calculating the Brown-York quasi-local energy on the domain
wall. Then the decay rate can be estimated by using the obtained parameters for
the inside Kerr geometry and the Brown-York quasi-local energy. Our results
support the statement that the BH spin suppresses the false vacuum decay in a
BH spacetime.
