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Inflation with $Ω\not = 1$

We discuss various models of inflationary universe with $Ω\not = 1$. A homogeneous universe with $Ω> 1$ may appear due to creation of the universe "from nothing" in the theories where the effective potential becomes very steep at large $ϕ$, or in the theories where the inflaton field $ϕ$ nonminimally couples to gravity. Inflation with $Ω< 1$ generally requires intermediate first order phase transition with the bubble formation, and with a second stage of inflation inside the bubble. It is possible to realize this scenario in the context of a theory of one scalar field, but typically it requires artificially bent effective potentials and/or nonminimal kinetic terms. It is much easier to obtain an open universe in the models involving two scalar fields. However, these models have their own specific problems. We propose three different models of this type which can describe an open homogeneous inflationary universe.