= Solution
Contract the <Einstein field equations> $G_{ab}=8\pi T_{ab}$ twice with $n^a$. Since $n^an_a=-1$ and $\rho=T_{ab}n^an^b$,
$$
R+2R_{ab}n^an^b=16\pi\rho.
$$
If $K_{ab}$ is proportional to $h_{ab}$ in three dimensions, tracing gives $K_{ab}=(K/3)h_{ab}$ and $K_{ab}K^{ab}=K^2/3$. The <Scalar Gauss equation> gives the <Hamiltonian constraint>
$$
{}^{(3)}R=16\pi\rho-\frac23K^2=\frac23(24\pi\rho-K^2).
$$
Therefore
$$
\boxed{24\pi\rho-K^2\geq0\quad\Longrightarrow\quad{}^{(3)}R\geq0}.
$$
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