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Instanton solutions mediating tunneling between the degenerate vacua in curved space

We investigate the instanton solution between the degenerate vacua in curved space. We show that there exist $O(4)$-symmetric solutions not only in de Sitter but also in both flat and anti-de Sitter space. The geometry of the new type of solutions is finite and preserves the $Z_2$ symmetry. The nontrivial solution corresponding to the tunneling is possible only if gravity is taken into account. The numerical solutions as well as the analytic computations using the thin-wall approximation are presented. We expect that these solutions do not have any negative mode as in the instanton solution.