Geometric interpretation for Einstein-scalar Lichnerowicz equation

Question

It is known that Einstein-scalar Lichnerowicz equation

$\Delta_gu-\frac{4(n-1)}{n-2}\Big(R_g-|\nabla\psi|_g^2\Big)u-\frac{4(n-1)}{n-2}\Big(Bu^{\frac{n+2}{n-2}}-Au^{-\frac{3n-2}{n-2}}\Big)=0.$

where $ R_g $ is scalar curvature and $\psi$ is scalar field.

It stems from the of Einstein constraint equations in general relativity. From the perspective of PDE, when $A=0$, this equation reduces to Yamabe-type equation. So that my question is whether there is a kind of geometric interpretation for this equation.

Thanks a lot.

Answer 1

The constraint equations are simply the expression for relationships that the metric and second fundamental form of a hypersurface inside a (Lorentzian) Einstein manifold must satisfy, according to the Gauss and Codazzi equations. So they are highly geometric (at least in the case of the vacuum Einstein constraints -- otherwise there is another term coming from the stress-energy tensor of the ambient manifold).