A general theory for local moduli space of minimal surface? 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. 
 A: In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark here (page 246):


Unfortunately minimal submanifolds
    are deﬁned by a second-order elliptic system and it is diﬃcult to understand the
    deformation theory. (Given a Jacobi ﬁeld on a minimal submanifold, can we ﬁnd a
    deformation by a family of minimal submanifolds along the ﬁeld?)


I think that http://arxiv.org/pdf/0709.1417v2.pdf provides an example of a branched minimal $S^2$ in $S^4$ with a non-integrable Jacobi field (see Theorem 4.1 and the subsequent comments). I'm not sure if there is a known example of a codimension one, embedded, minimal surface with non-integrable Jacobi fields. 

As I'm sure you know, the converse of your question is true: any "nearby" minimal surface corresponds to a Jacobi field. 
It depends on your exact problem, but you may get some mileage out of the "natural constraint," which allows you to associate a Jacobi field to a nearby surface which is minimal up to a finite dimensional error. This is described nicely in Leon Simon's book "Energy Minimizing Maps," Ch. 3.12 (or in many other places). 

See also http://www.ugr.es/~aros/icm-ros.pdf, Ch 7, for a discussion of the moduli space of minimal surfaces in $\mathbb{R}^3$ with finite total curvature. I think that it is not known whether or not this moduli space will be smooth (i.e. if non-integrable Jacobi fields exist).
It is known, however, that particular minimal surfaces have no non-integrable Jacobi fields. For example the Costa--Hoffman--Meeks surfaces of all genus have no non-integrable Jacobi fields: See http://arxiv.org/pdf/0806.1836.pdf. 

I'll also remark that your question is also related to the following question (of Yau, I think): Does there exist a $1$-parmeter family of non-isometric minimal surfaces in $\mathbb{S}^3$? An obvious strategy is to try to rule out non-trivial Jacobi fields, but this has not been successfully carried out. 
A: 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.
