Improved effective potential in the Einstein metric

We present a self-consistent evaluation of the effective potential for a λφ4 non-minimally coupled to an Einstein background geometry. This method takes into account the contribution from the multiloop Feynman diagrams with non-overlapping bubbles. It results that, even if the classical potential is double-well shaped, the vacuum expectation value of the scalar field is always vanishing because of a ''finite size'' effect, and no phase transitions can occur. A similar result can be obtained also in the de Sitter S4 space, where the present approach holds as well. The implications of the results on the dynamics of the early universe are briefly discussed. (author). 21 refs