# 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.

• I think Brian White has some results applying to moduli spaces of minimal surfaces: ams.org/mathscinet-getitem?mr=880951 ams.org/mathscinet-getitem?mr=1101226 Also, I think there are some powerful results of Colding and Minicozzi on spaces of minimal surfaces in $\mathbb{R}^3$. – Ian Agol Jun 6 '14 at 21:39
• John, you may find the second paper of White that @IanAgol linked particularly interesting, as it discusses conditions on $(N,g)$ so that the moduli space is smooth (see the Bumpy Metric Theorem on p 181). – Otis Chodosh Jun 7 '14 at 1:58

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.