Abstract
This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S n by looking at the conormal bundle of appropriate submanifolds of S n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in R n in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G2 and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anciaux, H.: Special Lagrangian submanifolds in the complex sphere, preprint, arXivimath.DG/0311288.
Atiyah, M. and Witten, E.: M-theory dynamics on a manifold of G2 holonomy, Adv. Theor. Math. Phys. 6 (2003), 1–106.
Bryant, R. L.: Conformal and minimal immersions of compact surfaces into the four-sphere, J. Differential Geom. 17 (1982), 455–473.
Bryant, R. L.: Submanifolds and special structures on the octonions, J. Differential Geom. 17 (1982), 185–232.
Bryant, R. L.: Metrics with exceptional holonomy, Ann. Math. 126 (1987), 525–576.
Bryant, R. L.: Some remarks on the geometry of austere manifolds, Bol. Soc. Brasil. Mat. (N.S.) 21 (1991), 133–157.
Bryant, R. L. and Salamon, S. M.: On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829–850.
Calabi, E.: Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. 12(4) (1979), 269–294.
Cvetič, M., Gibbons, G. W., Lü, H. and Pope, C. N.: Ricci-flat metrics, harmonic forms, and brane resolutions, Comm. Math. Phys. 232 (2003), 457–500.
Cvetič, M., Gibbons, G. W., Lü, H. and Pope, C. N.: Hyper-Kähler calabi metrics, L 2 harmonic forms, resolved M2-branes, and AdS 4/CFT 3 correspondence, Nuclear Phys. B 617 (2001), 151–197.
Dajczer, M. and Florit, L. A.: A class of austere submanifolds, Illinois J. Math. 45 (2001), 735–755.
Dancer, A. and Strachan, I. A. B.: Einstein metrics on tangent bundles of spheres, Classical Quantum Gravity 19 (2002), 4663–4670.
Dancer, A. and Swann, A.: Hyperkähler metrics of cohomogeneity one, J. Geom. Phys. 21 (1997), 218–230.
Eells, J. and Salamon, S.: Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 12 (1985), 589–640.
Gibbons, G. W., Page, D. N. and Pope, C. N. Einstein metrics on S 3, ℝ3, and ℝ4 bundles, Comm. Math. Phys. 127 (1990), 529–553.
Gukov S. and Sparks, J.: M-Theory on spin(7) manifolds, Nuclear Phys. B 625 (2002), 3–69.
Harvey, R. Spinors and Calibrations, Academic Press, San Diego, 1990.
Harvey, R. and Lawson, H. B.: A constellation of minimal varieties defined over the group G 2, lecture notes, Appl. Math. 48 (1979), 167–187.
Harvey, R. and Lawson, H. B.: Calibrated geometries, Acta Math. 148 (1982), 47–157.
Hitchin, N.: The geometry of three-forms in six and dimensions, J. Differential Geom. 55 (2000), 547–576.
Ionel, M., Karigiannis, S. and Min-Oo, M.: Bundle constructions of calibrated submanifolds in ℝ7 and ℝ8, Math. Res. Lett. 12 (2005), 493–512.
Joyce, D. D.: Compact Manifolds with Special Holonomy, Oxford University Press, Oxford, 2000.
Joyce, D. D.: Lectures on special Lagrangian geometry, arXiv: math.DG/0111111.
Joyce, D. D.: The exceptional holonomy groups and calibrated geometry, arXiv: math.DG/0406011.
Joyce, D. D.: Special lagrangian submanifolds with isolated conical singularities. II. moduli spaces, Ann. Global Anal. Geom. 25 (2004), 303–352.
Joyce, D. D. and Salur, S.: Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary, arXiv: math.DG/0408137.
Karigiannis, S.: Deformations of G 2 and spin(7) structures, Canad. J. Math. 57 (2005), 1012–1055.
Karigiannis, S.: A note on signs and orientations in G 2 and Spin(7) geometry, posted on the arXiv.
Lee, J. H. and Leung, N. C.: Instantons and branes in manifolds with vector cross product, arXiv: math.DG/0402044.
Leung, N. C. and Wang, X. W.: Intersection theory of coassociative submanifolds in G 2 manifolds and Seiberg–Witten invariants, arXiv: math.DG/0401419.
Lotay, J.: Constructing associative 3-folds by evolution equations, arXiv: math.DG/0401123.
Lotay, J.: 2-Ruled calibrated 4-folds in ℝ7 and ℝ8, arXiv: math.DG/0401125.
Lotay, J.: Deformation theory of asymptotically conical coassociative 4-folds, arXiv: math.DG/0411116.
Marshall, S. P.: Deformations of special Lagrangian submanifolds, Oxford DPhil. Thesis 2002.
McLean, R. C.: Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747.
Pacini, T.: Deformations of asymptotically conical special Lagrangian submanifolds, Pacific J. Math. in press.
Salamon, S. M.: Harmonic and holomorphic maps, Lecture Notes in Math. 1164, Springer, New York, 1984, pp. 161–224.
Salamon, S. M.: Riemannian Geometry and Holonomy Groups, Longman, Harlow, 1989.
Stenzel, M. B.: Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80 (1993), 151–163.
Strominger, A., Yau, S. T. and Zaslow, E.: Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243–259.
Szöke, R.: Complex structures on tangent bundles of eiemannian manifolds, Math. Ann. 291 (1991), 409–428.
Wang, S. H.: On the lifts of minimal Lagrangian submanifolds, arXiv: math.DG/0109214.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 53-XX, 58-XX.
Rights and permissions
About this article
Cite this article
Karigiannis, S., Min-Oo, M. Calibrated Subbundles in Noncompact Manifolds of Special Holonomy. Ann Glob Anal Geom 28, 371–394 (2005). https://doi.org/10.1007/s10455-005-1940-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10455-005-1940-7