Calibrated embeddings in the special Lagrangian and coassociative cases

TL;DR

This paper demonstrates that all closed, oriented, real analytic 3- and 4-manifolds can be isometrically embedded as special Lagrangian or coassociative submanifolds in Calabi-Yau and G_2-manifolds, respectively, expanding the known examples of such calibrated geometries.

Contribution

It provides new embedding results for real analytic 3- and 4-manifolds into special Lagrangian and coassociative submanifolds, respectively, in Calabi-Yau and G_2-manifolds.

Findings

Every real analytic 3-manifold can be embedded as a special Lagrangian submanifold.

Every real analytic 4-manifold with trivial self-dual bundle can be embedded as a coassociative submanifold.

These embeddings produce examples with nontrivial deformation spaces.

Abstract

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.