N-dimensional static and evolving Lorentzian wormholes with a cosmological constant

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We present a family of static and evolving spherically symmetric Lorentzian wormhole solutions in N+1 dimensional Einstein gravity. In general, for static wormholes, we require that at least the radial pressure has a barotropic equation of state of the form $p_r=\omega_r \rho$, where the state parameter $\omega_r$ is constant. On the other hand, it is shown that in any dimension $N \geq 3$, with $\phi(r)=\Lambda=0$ and anisotropic barotropic pressure with constant state parameters, static wormhole configurations are always asymptotically flat spacetimes, while in 2+1 gravity there are not only asymptotically flat static wormholes and also more general ones. In this case, the matter sustaining the three-dimensional wormhole may be only a pressureless fluid. In the case of evolving wormholes with $N \geq 3$, the presence of a cosmological constant leads to an expansion or contraction of the wormhole configurations: for positive cosmological constant we have wormholes which expand forever and, for negative cosmological constant we have wormholes which expand to a maximum value and then recollapse. In the absence of a cosmological constant the wormhole expands with constant velocity, i.e without acceleration or deceleration. In 2+1 dimensions the expanding wormholes always have an isotropic and homogeneous pressure, depending only on the time coordinate.