It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point, and any two othogonal vectors in this plane, and any negative number $K$ there exists a (in fact, infinitely many) minimal suface tangent to the plane at that point such that main curvature vectors are that vectors and the gauss curvature is that number.
Is the statement remains true if our $R^3$ is equipped by a not necessary flat metric?
On the level of equations, the system is of course very underdetermined so one expects a similar freedom to that in the standard flat situation.
The motivation is due to a certain question asked after my talk on projective equivalence of metrics (whether it is reasonable to replace geodesics by minimal surfaces in the definition of projective equivalence); would the answer to the question be positive, there is no sense of doing it since two different metrics in $R^3$ such that each minimal surface of the first is a minimal surface of the second will be proportional with a constant coefficient.
Of course the natural generalisation(s) of this question is also interesting in higher dimensions.