are there any examples of a real analytic riemannian manifold that cannot be isometrically embedded as a special lagrangian submanifold of a calabiyau manifold ?
peter hara
are there any examples of a real analytic riemannian manifold that cannot be isometrically embedded as a special lagrangian submanifold of a calabiyau manifold ? peter hara 


For the first question, just note that, already for dimension 2, the space of compact CalabiYau surfaces is a finitedimensional space, and the metrics that can be realized on compact complex curves in such a CalabiYau fall into a countable union of finite dimensional families. (Remember that special Lagrangian surfaces in a CalabiYau are complex curves in a different CalabiYau metric in the canonical $S^2$family of CalabiYau metrics.) Thus, the set of such realizable metrics, even on the $2$sphere, constitutes a countable union of finite dimensional families. This could never account for all of the realanalytic metrics on the $2$sphere. Thus, some example exists, though we don't know one explicitly. For the second question, consider the fact that it is highly unlikely that the induced metric on any complex curve in a CalabiYau surface has constant Gaussian curvature. The 'reason' is that most (nonflat) Ricciflat Kahler metrics contain no complex curves with constant Gaussian curvature. It would be remarkable indeed if one of the Ricciflat Kahler metrics on a (nonflat) compact 4manifold had such a curve. In particular, I regard it as highly likely that the standard round metric on the $2$sphere cannot be isometrically embedded as a complex curve in any compact CalabiYau surface. My answer to the third question is just an affirmation of my ignorance. A remark about the local story: peter h asked about what I would call the 'local case', i.e., whether a real analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in some CalabiYau, with no assumptions about completeness of the ambient manifold. In particular, he raised the question for surfaces. Now, in the case of a realanalytic metric on a Riemann surface, the answer would be 'yes', according to a paper in 2000 by D. Kaledin, "Hyperkaehler structures on total spaces of holomorphic cotangent bundles", which is available on the arXive (arXiv:alggeom/9710026v1). (It's 100 pages, and I don't claim that I have read it, I'm just pointing out that it is there.) The main theorem of this paper is that, given any realanalytic Kahler manifold $M$, there exists a hyperKahler metric on a neighborhood of the $0$section of the cotangent bundle $T^\ast M$ that is compatible with the natural complex and holomorphic structures on $T^\ast M$ and that induces the original metric on the $0$section. When the (real) dimension of $M$ is $2$, this would apply to show that $M$ is isometrically imbedded as a complex curve in a CalabiYau (complex) surface, and then one can apply the 'rotation trick' to turn this into a special Lagrangian surface when the ambient $4$manifold is regarded as a complex surface with respect to one of the orthogonal complex structures. Thus, the case of surfaces would be covered by this theorem. In fact, this would work in any even dimension when the given realanalytic metric is actually Kahler. There would remain the question (which I raised in my original paper) of whether every realanalytic metric on $S^4$ can be realized by an embedding as a special Lagrangian submanifold of a $4$dimensional CalabiYau. 


On the contrary, R. Bryant has shown that any closed oriented real analytic 3dimensional riemannian manifold is the real locus of an antiholomorphic, isometric involution of a CalabiYau 3fold (see http://arxiv.org/abs/math/9912246). 

