##
**Calibrated geometries.**
*(English)*
Zbl 0584.53021

This paper is perhaps best characterized as a foundational essay on the geometries of minimal varieties associated to closed forms. The fundamental observation here is the following: Let \(X\) be a Riemannian manifold, and suppose \(\phi\) is a closed exterior p-form with the property that
\[
(1)\quad \phi | \xi \leq vol_{\xi}
\]
for all oriented tangent \(p\)-planes \(\xi\) on \(X\). Then any compact oriented \(p\)- dimensional submanifold \(M\) of \(X\) with the property that
\[
(2)\quad \phi |_ M=vol_ M
\]
is homologically volume minimizing in \(X\), i.e. vol(M)\(\leq vol(M')\) for any \(M'\) such that \(\partial M=\partial M'\) and \([M-M']=0\) in \(H_ p(X;R)\). To see this, one simply notes that \(vol(M)=\int_ M\phi =\int_{M'}\phi \leq vol(M')\). Condition (2) enables us to associate to an exterior \(p\)-form \(\phi\) a family of oriented \(p\)-dimensional submanifolds in \(X\) which we call \(\phi\)-submanifolds. If \(\phi\) is closed and is normalized to satisfy condition (1), then each \(\phi\)-submanifold is homologically mass minimizing in \(X\).

A closed exterior \(p\)-form \(\phi\) satisfying (1) will be called a calibration and the Riemannian manifold \(X\) together with this form will be called a calibrated manifold. As an example, let \(X\) be a complex Hermitian \(n\)-manifold with Kähler form \(\omega\), and consider \(\phi =(1/p!)\omega^ p\) for some \(p\), \(1\leq p\leq n\). Then the \(\phi\)- submanifolds are just the canonically oriented complex submanifolds of dimension \(p\) in \(X\). If \(d\phi =0\), i.e., if \(X\) is a Kähler manifold, then the complex submanifolds are homologically mass minimizing. This is the classical observation of H. Federer [Geometric measure theory. Berlin: Springer (1969; Zbl 0176.00801)]. One of the main points of this paper is to exhibit and study some beautiful geometries of minimal subvarieties which are really not visible from this first viewpoint. We shall concentrate primarily on geometries in \(\mathbb R^ n\) associated to forms with constant coefficients. A significant part of the work will be to derive a tractable system of partial differential equations whose solutions represent subvarieties in the given geometry. These systems are in a specific sense generalizations of the Cauchy-Riemann equations.

The first geometry to be studied in depth is associated to the form \[ \phi =\operatorname{Re}\{dz_ 1\bigwedge...\bigwedge\, dz_ n\} \] in \(\mathbb C^ n\). It consists of Lagrangian submanifolds of ”constant phase”, and is therefore called special Lagrangian geometry. In fact the only Lagrangian submanifolds which are stationary are special Lagrangian.

Up to \(SU_ n\)-coordinate changes, special Lagrangian submanifolds are locally graphs of the form \(\{y=(\nabla F)(x)\}\) where \(F\) is a scalar potential function satisfying a nonlinear elliptic equation. When \(n=3\), this equation has the following beautiful form: \[ (3)\quad \Delta F=\det (\text{Hess }F). \] We conclude that the graph of the gradient of any solution to (3) is an absolutely volume-minimizing three-fold in \(\mathbb R^ 6\). In particular, any \(C^ 2\) solution of (3) is real analytic. The equation (3) bears an intimate relation to the work of H. Lewy on harmonic gradient maps [Ann. Math. (2) 88, 518–529 (1968; Zbl 0164.13803)] and explains the mysterious appearance there of the minimal surface equation. This is discussed in Chapter III.

The geometry of special Lagrangian submanifolds in richly endowed (see Sections III.3 and 4), and constitutes a large new class of minimizing currents in \(\mathbb R^ n\). In particular, we are able to explicitly construct simple minimizing cones which are not real analytic (see Section III.3.C).

Chapter IV is devoted to the study of three exceptional geometries. There is a geometry of three-folds (and a dual geometry of four-folds) in \(\mathbb R^ 7\), which is invariant under the standard representation of \(G_ 2\). This geometry is associated to the three-form \(\phi (x,y,z)=(x,yz)\) where \(x,y,z\in \mathbb R^ 7\) are considered as imaginary Cayley numbers. A three- manifold \(M\subset \mathbb R^ 7=\text{Im }O\) belongs to this geometry if each of its tangent planes is a (canonically oriented) imaginary part of a quaternion subalgebra of the Cayley numbers \(O\). The local system of differential equations for this geometry is essentially deduced from the vanishing of the associator \([x,y,z]=(xy)z-x(yz)\), and thus the geometry is called associative. The most fascinating and complex geometry discussed here is the geometry of Cayley four-folds in \(\mathbb R^ 8\cong O\). This is the family of subvarieties associated to the four-form \(\psi (x,y,z,w)=\langle x(\bar yz)- z(\bar yx),w\rangle.\) It is invariant under the eight-dimensional representation of \(\text{Spin}_ 7\) and contains the coassociative geometry (the dual geometry of four-folds in \({\mathbb R}^ 7)\). It also contains both the (negatively oriented) complex and the special Lagrangian geometries for a seven-dimensional family of complex structures on \({\mathbb{R}}^ 8\). In fact for any of these structures, the form \(\psi\) can be expressed as \[ \psi =-1/2\omega^ 2+\operatorname{Re}\{dz\} \] where \(\omega\) is the Kähler form and \(dz=dz_ 1\bigwedge...\bigwedge\, dz_ 4\) as above.

Chapter V contains a number of comments concerning generalizations of the main ideas and results of the paper. These comments include the observation that every Cayley four-fold naturally carries a twentyone- dimensional family of anti-self-dual \(SU_ 2\) Yang-Mills fields.

A closed exterior \(p\)-form \(\phi\) satisfying (1) will be called a calibration and the Riemannian manifold \(X\) together with this form will be called a calibrated manifold. As an example, let \(X\) be a complex Hermitian \(n\)-manifold with Kähler form \(\omega\), and consider \(\phi =(1/p!)\omega^ p\) for some \(p\), \(1\leq p\leq n\). Then the \(\phi\)- submanifolds are just the canonically oriented complex submanifolds of dimension \(p\) in \(X\). If \(d\phi =0\), i.e., if \(X\) is a Kähler manifold, then the complex submanifolds are homologically mass minimizing. This is the classical observation of H. Federer [Geometric measure theory. Berlin: Springer (1969; Zbl 0176.00801)]. One of the main points of this paper is to exhibit and study some beautiful geometries of minimal subvarieties which are really not visible from this first viewpoint. We shall concentrate primarily on geometries in \(\mathbb R^ n\) associated to forms with constant coefficients. A significant part of the work will be to derive a tractable system of partial differential equations whose solutions represent subvarieties in the given geometry. These systems are in a specific sense generalizations of the Cauchy-Riemann equations.

The first geometry to be studied in depth is associated to the form \[ \phi =\operatorname{Re}\{dz_ 1\bigwedge...\bigwedge\, dz_ n\} \] in \(\mathbb C^ n\). It consists of Lagrangian submanifolds of ”constant phase”, and is therefore called special Lagrangian geometry. In fact the only Lagrangian submanifolds which are stationary are special Lagrangian.

Up to \(SU_ n\)-coordinate changes, special Lagrangian submanifolds are locally graphs of the form \(\{y=(\nabla F)(x)\}\) where \(F\) is a scalar potential function satisfying a nonlinear elliptic equation. When \(n=3\), this equation has the following beautiful form: \[ (3)\quad \Delta F=\det (\text{Hess }F). \] We conclude that the graph of the gradient of any solution to (3) is an absolutely volume-minimizing three-fold in \(\mathbb R^ 6\). In particular, any \(C^ 2\) solution of (3) is real analytic. The equation (3) bears an intimate relation to the work of H. Lewy on harmonic gradient maps [Ann. Math. (2) 88, 518–529 (1968; Zbl 0164.13803)] and explains the mysterious appearance there of the minimal surface equation. This is discussed in Chapter III.

The geometry of special Lagrangian submanifolds in richly endowed (see Sections III.3 and 4), and constitutes a large new class of minimizing currents in \(\mathbb R^ n\). In particular, we are able to explicitly construct simple minimizing cones which are not real analytic (see Section III.3.C).

Chapter IV is devoted to the study of three exceptional geometries. There is a geometry of three-folds (and a dual geometry of four-folds) in \(\mathbb R^ 7\), which is invariant under the standard representation of \(G_ 2\). This geometry is associated to the three-form \(\phi (x,y,z)=(x,yz)\) where \(x,y,z\in \mathbb R^ 7\) are considered as imaginary Cayley numbers. A three- manifold \(M\subset \mathbb R^ 7=\text{Im }O\) belongs to this geometry if each of its tangent planes is a (canonically oriented) imaginary part of a quaternion subalgebra of the Cayley numbers \(O\). The local system of differential equations for this geometry is essentially deduced from the vanishing of the associator \([x,y,z]=(xy)z-x(yz)\), and thus the geometry is called associative. The most fascinating and complex geometry discussed here is the geometry of Cayley four-folds in \(\mathbb R^ 8\cong O\). This is the family of subvarieties associated to the four-form \(\psi (x,y,z,w)=\langle x(\bar yz)- z(\bar yx),w\rangle.\) It is invariant under the eight-dimensional representation of \(\text{Spin}_ 7\) and contains the coassociative geometry (the dual geometry of four-folds in \({\mathbb R}^ 7)\). It also contains both the (negatively oriented) complex and the special Lagrangian geometries for a seven-dimensional family of complex structures on \({\mathbb{R}}^ 8\). In fact for any of these structures, the form \(\psi\) can be expressed as \[ \psi =-1/2\omega^ 2+\operatorname{Re}\{dz\} \] where \(\omega\) is the Kähler form and \(dz=dz_ 1\bigwedge...\bigwedge\, dz_ 4\) as above.

Chapter V contains a number of comments concerning generalizations of the main ideas and results of the paper. These comments include the observation that every Cayley four-fold naturally carries a twentyone- dimensional family of anti-self-dual \(SU_ 2\) Yang-Mills fields.

### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58A25 | Currents in global analysis |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

32C30 | Integration on analytic sets and spaces, currents |

32Q99 | Complex manifolds |

### Keywords:

minimal varieties; closed forms; \(\phi \) -submanifolds; homologically mass minimizing; calibrated manifold; Kähler manifold; Cauchy-Riemann equations; Lagrangian submanifolds; special Lagrangian geometry; minimizing currents; minimizing cones; exceptional geometries; Cayley numbers; coassociative geometry; complex structures; Kähler form; Yang-Mills fields
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\textit{R. Harvey} and \textit{H. B. Lawson}, Acta Math. 148, 47--157 (1982; Zbl 0584.53021)

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