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As Dickfore points out, the 3D Einstein equation has a completely different physical character than the 4D one. All degrees of freedom in the curvature are locally determined by the matter distribution. This means, for example, that there can be no gravitational radiation. And there can be no analog of the Schwarzschild solution.
What might be more interesting is to look at a 2+1 slice of a static, vacuum, cylindrically symmetric 3+1 solution. This describes the Levi-Civita metric:
[tex] ds^2 = - \rho^{4 \alpha} d t^2 + \rho^{4 \alpha (2 \alpha-1)} ( d \rho^2 + d z^2 ) + \beta^{-2} \rho^{2-4\alpha} d \phi^2 .[/tex]
[itex]\alpha[/itex] can be interpreted as a mass/length and [itex]\beta[/itex] is an angular defect parameter.
What might be more interesting is to look at a 2+1 slice of a static, vacuum, cylindrically symmetric 3+1 solution. This describes the Levi-Civita metric:
[tex] ds^2 = - \rho^{4 \alpha} d t^2 + \rho^{4 \alpha (2 \alpha-1)} ( d \rho^2 + d z^2 ) + \beta^{-2} \rho^{2-4\alpha} d \phi^2 .[/tex]
[itex]\alpha[/itex] can be interpreted as a mass/length and [itex]\beta[/itex] is an angular defect parameter.