A general theory for local moduli space of minimal surface?

Question

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.

Answer 1

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.

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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 "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.

Answer 2

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.