Let $M$ be a closed Riemannian manifold. I have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.

(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point of the area functional. Considering the 2nd variation, one has the Jacobi operator. So the kernel of the Jacobi operator $K$ corresponds to the first order deformation of minimal submanifolds near $N$. Will there be a map

$$ K \to \{\text{all minimal submanifolds in }M\}$$

Such that it is locally surjective around $N$?

(2) What can we say about the structure of the set of all minimal submanifolds in $M$?

If the question is too board, please feel free to impose restrictions (for example, dimension on $M$, $N$ or even some specific $M$)

**Remark:** In some very special case (for example special Lagrangian submanifolds in a Calabi-Yau $M$), $K$ (need to restrict to Lagrangian variation here) actually parametrizes locally all nearby special Lagrangians. But the proof uses the fact that all special Lagrangians are calibrated, so cannot be generalized.