##
**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
PDFBibTeX
XMLCite

\textit{R. Harvey} and \textit{H. B. Lawson}, Acta Math. 148, 47--157 (1982; Zbl 0584.53021)

Full Text:
DOI

### References:

[1] | Arnold, V. I.,Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1978. · Zbl 0386.70001 |

[2] | Berger, M., Sur les groups d’holonomie homogene des varieties à connexion affine et des varietie riemanniennes.Bull. Soc. Math. France, 83 (1955), 279–330. · Zbl 0068.36002 |

[3] | –, Du côté de chez pu.Ann. Sci. École Norm. Sup., 5 (1972), 1–44. · Zbl 0227.52005 |

[4] | Bombieri, E., Algebraic values of meromorphic maps.Invent. Math., 10 (1970), 267–287. · Zbl 0214.33702 · doi:10.1007/BF01418775 |

[5] | Bryant, R. L., Submanifolds and special structures on the octonions.J. Differential Geom., 17 (1982). · Zbl 0526.53055 |

[6] | Brown, R. &Gray, A., Vector cross products.Comment. Math. Helv., 42 (1967), 222–236. · Zbl 0155.35702 · doi:10.1007/BF02564418 |

[7] | Calabi, E., Construction and some properties of some 6-dimensional almost complex manifolds.Trans. Amer. Math. Soc., 87 (1958), 407–438. · Zbl 0080.37601 |

[8] | Curtis, C. W., The four and eight square problem and division algebras, inStudies in modern algebra, Vol. 2 MAA Studies in Mathematics (1963), 100–125. |

[9] | Do Carmo M. &Wallach, N. R., Representations of compact groups and minimal immersions into spheres.J. Differential Geom., 4 (1970), 91–104. · Zbl 0197.18301 |

[10] | Federer, H.,Geometric Measure Theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801 |

[11] | –, Real flat chains, cochains and variational problems.Indiana Univ. Math. J., 24 (1974), 351–407. · Zbl 0289.49044 · doi:10.1512/iumj.1974.24.24031 |

[12] | –, Colloquium lectures on geometric measure theory.Bull. Amer. Math. Soc., 84 (1978), 291–338. · Zbl 0392.49021 · doi:10.1090/S0002-9904-1978-14462-0 |

[13] | Federer, H. &Fleming, W. H., Normal and integral currents.Ann. of Math., 72 (1960), 458–520. · Zbl 0187.31301 · doi:10.2307/1970227 |

[14] | Gray, A., Vector cross products on manifolds.Trans. Amer. Math. Soc., 141 (1969), 465–504. · Zbl 0182.24603 · doi:10.1090/S0002-9947-1969-0243469-5 |

[15] | Harvey, R., Removable singularities for positive currents.Amer. J. Math., 96 (1974), 67–78. · Zbl 0293.32015 · doi:10.2307/2373581 |

[16] | –, Holomorphic chains and their boundaries,Proceedings of Symposia in Pure Mathematics, 30. A.M.S., Providence, R.I. (1977), 309–382. |

[17] | Harvey, R. &King, J., On the structure of positive currents.Invent. Math., 15 (1972), 47–52. · Zbl 0235.32006 · doi:10.1007/BF01418641 |

[18] | Harvey, R. &Knapp, A. W., Positive (p, p)-forms, Wirtinger’s inequality and currents, value distribution theory, Part A.Proceedings Tulane University. Program on Value-Distribution Theory in Complex Analysis and Related Topics in Differential Geometry, 1972–1973, pp. 43–62. Dekker, New York, 1974. |

[19] | Harvey, R. &Lawson, Jr., H. B., On boundaries of complex analytic varieties, I.Ann. of Math., 102 (1975), 233–290. · Zbl 0317.32017 · doi:10.2307/1971032 |

[20] | –, On boundaries of complex analytic varieties, II.Ann. of Math., 106 (1977), 213–238. · Zbl 0361.32010 · doi:10.2307/1971093 |

[21] | Harvey, R., A constellation of minimal varieties defined over the groupG 2. Proceedings of Conference on Geometry and Partial Differential Equations,Lecture Notes in Pure and Applied Mathematics, 48. Marcel-Dekker (1979), 167–187. |

[22] | –, Geometries associated to the groupSU n and varieties of minimal submanifolds arising from the Cayley arithmetic, inMinimal submanifolds and geodesics. Kaigai Publications, Tokyo (1978), 43–59. |

[23] | –, Calibrated foliations.Amer. J. Math., 103 (1981), 411–435. · Zbl 0538.57006 · doi:10.2307/2374099 |

[24] | Harvey, R. &Polking, J., Removable singularities of solutions of linear partial differential equations.Acta Math., 125 (1970), 39–56. · Zbl 0214.10001 · doi:10.1007/BF02838327 |

[25] | –, Extending analytic objects.Comm. Pure Appl. Math., 28 (1975), 701–727. · Zbl 0323.32013 · doi:10.1002/cpa.3160280603 |

[26] | Harvey, R. &Shiffman, B., A characterization of holomorphic chains.Ann. of Math., 99 (1974), 553–587. · Zbl 0287.32008 · doi:10.2307/1971062 |

[27] | Helgason, S.,Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978. · Zbl 0451.53038 |

[28] | Hsiang, W. Y. &Lawson, Jr., H. B., Minimal submanifolds of low cohomogeneity.J. Differential Geom., 5 (1971), 1–38. · Zbl 0219.53045 |

[29] | King, J., The currents defined by analytic varieties.Acta Math., 127 (1971), 185–220. · Zbl 0224.32008 · doi:10.1007/BF02392053 |

[30] | Kleinfeld, E., A characterization of the Cayley numbers, inStudies in modern algebra, Vol. 2 MAA Studies in Mathematics (1963), 126–143. |

[31] | Kobayashi, S. &Nomizu, K.,Foundations of Differential Geometry, Iand II. Wiley-Interscience, New York, 1963 and 1969. · Zbl 0119.37502 |

[32] | Lawson, Jr., H. B., Complete minimal surfaces inS 3.Ann. of Math., 92 (1970), 335–374. · Zbl 0205.52001 · doi:10.2307/1970625 |

[33] | –, The equivariant Plateau problem and interior regularity.Trans. Amer. Math. Soc., 173 (1972), 231–250. · Zbl 0279.49043 · doi:10.1090/S0002-9947-1972-0308905-4 |

[34] | –, Foliations.Bull. Amer. Math. Soc., 80 (1974), 369–417. · Zbl 0293.57014 · doi:10.1090/S0002-9904-1974-13432-4 |

[35] | –,Minimal varieties in real and complex geometry. University of Montreal Press, Montreal 1973. |

[36] | Lawson, Jr., H. B. &Osserman, R., Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system.Acta Math., 139 (1977), 1–17. · Zbl 0376.49016 · doi:10.1007/BF02392232 |

[37] | Lelong, P.,Fonctions plurisousharmoniques et formes differentielles positives. Gordon and Breach, New York; distributed by Dunod Éditeur, Paris, 1968. · Zbl 0195.11603 |

[38] | Lelong, P., Sur la structure des courants positifs fermés. Séminaire Pierre Lelong,Lecture Notes in Mathematics, 578. Springer (1975/76), 136–156. |

[39] | Lewy, H., On the non-vanishing of the Jacobian of a homeomorphism by harmonic gradients,Ann. of Math., 88 (1968), 578–529. · Zbl 0164.13803 · doi:10.2307/1970723 |

[40] | Morrey, C. B., Second order elliptic systems of partial differential equations, pp. 101–160, in Contributions to the Theory of Partial Differential Equations.Ann. of Math. Studies, 33, Princeton Univ. Press, Princeton, 1954. · Zbl 0057.08301 |

[41] | Ruelle, D. &Sullivan, D., Currents, flows and diffeomorphisms.Topology, 14 (1975), 319–327. · Zbl 0321.58019 · doi:10.1016/0040-9383(75)90016-6 |

[42] | Simons, J., On transitivity holonomy systems.Ann of Math., 76 (1962), 213–234. · Zbl 0106.15201 · doi:10.2307/1970273 |

[43] | Siu, Y. T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents.Invent. Math., 27 (1974), 53–156. · Zbl 0289.32003 · doi:10.1007/BF01389965 |

[44] | Spivak, M.,Differential geometry, V. Publish or Perish, Berkeley 1979. |

[45] | Yau, S. T., Calabi’s conjecture and some new results in algebraic geometry,Proc. Nat. Acad. Sci. U.S.A., 74 (1977), 1798–1799. · Zbl 0355.32028 · doi:10.1073/pnas.74.5.1798 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.