The question is not precisely posed. The equations that govern the curvature are got by varying the metric components as dynamical variables in an action that comprises the integral of the curvature scalar ( which is a function of the metric and its derivatives).
In 4-D space-time one can define 3D and 2D 'slices' which may have interesting properties, but I don't know if these properties can be varied to give the correct field equations.
Ok thanks for replying. I somehow recall seeing something about this. As you guys know, Quantum Mechanics is based in the space of complex analytic functions and mimimal surfaces are described by the same sort of functions and I was wondering if this common link of complex analytic functions connect the very small and very large. I'm not however familiar with General Relativity and do now know how such functions participate in the general theory. Might some of you comment further about this connection?
As I understand it, a "minimal surface" is a surface that minimizes surface area subject to boundary constraints. In order for this notion to be well-defined, one needs (1) a boundary, and (2) a precise way of performing variational calculus on the surface in order to extremize the surface area functional. Both of these conditions are absent in the context of general relativity. In fact, spaces that possesses property (1), i.e., have a "boundary" in some sense, often appear in general relativity, but only as incomplete sections of larger manifolds. More precisely, if paths on the manifold "end" at finite values of their parameters, then the space is said to be geodesically incomplete, and great effort is invested in finding new coordinate systems that fix this problem.
The second condition is a bit more subtle. Without going into details, it can be shown that if (2) is satisfied, then the condition for minimality is that the mean curvature identically vanish. Depending on your familiarity with surface theory, you may or may not know that the mean curvature is also referred to as the "extrinsic" curvature (as opposed to the Gaussian, or "intrinsic," curvature), because it is dependent on the particular embedding chosen (rather than depending entirely on the metric structure of the surface, which is what "intrinsic" means). In other words, two surfaces that are isometric (i.e., there exists a smooth map between them that preserves the lengths of curves) need not have the same mean curvature. In GR, the spaces of interest are completely devoid of any embedding in some higher-dimensional space; the only properties that matter are the intrinsic, metric-dependent ones. Thus, the statement that the mean curvature vanish identically is completely meaningless in GR. This is not to say that spacetime manifolds can't be embedded in some higher-dimensional manifold (there is a theorem that states that it is always possible to embed an n-dimensional manifold in 2n-dimensional Euclidean space), but rather that GR does not specify a way to do this, so Einstein's equations have nothing to say about the extrinsic curvature of spacetime with respect to any particular embedding you might choose.