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 general theory for local moduli space of minimal surface?
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](http://faculty.ccri.edu/joallen/M2910/Geometry%20and%20Analysis_Yau.pdf) (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. 




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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](http://www.ams.org/mathscinet-getitem?mr=1399562) "Energy Minimizing Maps," Ch. 3.12 (or in many other places). 




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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>. 




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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. 


