TITLE: A general theory for local moduli space of minimal surface?
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Although this thread has been long been inactive, I want to add an answer that I hope will complement that already given by Otis Chodosh. Specifically I want to address the comment that Otis made in the second paragraph, where he asks for an 'example of a codimension one, embedded, minimal surface with non-integrable Jacobi fields.'
Let $M^2 \subset \mathbf{R}^3$ be the surface given by the equation $x^2 + y^2 + y^4 = 1$, and let $\Sigma = M \cap \{ z = 0 \}$ be the geodesic obtained by intersecting it with the horizontal plane. The Gauss curvature $K$ of $M$ is strictly positive everywhere except on $\Sigma$, where it is identically zero.
The vector field $\partial / \partial z$ is a Jacobi field of $\Sigma$, but is not integrable because of the Gauss-Bonnet theorem. Indeed, if $(\Sigma_t)$ were a family of geodesics with $\Sigma_0 = \Sigma$ generating $\partial / \partial z$ then for $t$ small enough, $\Sigma_t$ would lie strictly above $\Sigma$, but not intersect it. Let $U \subset M$ be the region bounded by $\Sigma$ and such a $\Sigma_t$. Then $0 < \int_U K = 2 \pi \chi(U) = 0$, which is patently absurd.
By taking products of $M$ with $\mathbf{S}^1$ one obtains examples of higher dimension: for example $\Sigma \times \mathbf{S}^1 \subset M \times \mathbf{S}^1 \subset \mathbf{R}^4$ is a surface with a non-integrable Jacobi field. This example is taken from a paper by Leung [1], who attributes it to Almgren.
[1] D.S.P. Leung. On the Integrability of Jacobi Fields on Minimal Submanifolds. Transactions of the American Mathematical Society. Vol. 220 (1976), pp. 185-194.