Quaternionic maps and minimal surfaces. (English) Zbl 1170.53312
Summary: Let \((M,J^\alpha,\alpha=1,2,3)\) and \((N,{\mathcal J}^\alpha,\alpha= 1,2,3)\) be hyperkähler manifolds. We study stationary quaternionic maps between \(M\) and \(N\). We first show that if there are no holomorphic 2-spheres in the target then any sequence of stationary quaternionic maps with bounded energy subconverges to a stationary quaternionic map strongly in \(W^{1,2}(M,N)\). We then find that certain tangent maps of quaternionic maps give rise to an interesting minimal 2-sphere. At last we construct a stationary quaternionic map with a codimension-3 singular set by using the embedded minimal \(\mathbb{S}^2\) in the hyperkähler surface \(\widetilde M^0_2\) studied by Atiyah-Hitchin.
MSC:
| 53C43 | Differential geometric aspects of harmonic maps |
| 58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |
| 58E20 | Harmonic maps, etc. |
References:
| [1] | D. V. Alekseevsky and S. Marchiafava, A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold, Ann. Mat. Pura Appl. 184 (2005), 53-74. MR2128094 |
| [2] | D. Anselmi and P. Fré, Topological Sigma-Models in Four Dimensions and Triholomorphic Maps, Nucl. Phys. B416 (1994), 255-300. Zbl1007.53500 MR1272647 · Zbl 1007.53500 · doi:10.1016/0550-3213(94)90585-1 |
| [3] | M. Atiyah and N. Hitchin, “The geometry and dynamics of magnetic monopoles”, Princeton University Press 1988. Zbl0671.53001 MR934202 · Zbl 0671.53001 |
| [4] | F. Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), 417-443. Zbl0792.53039 MR1208652 · Zbl 0792.53039 · doi:10.1007/BF02599324 |
| [5] | J. Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic curves, Comm. Math. Phys. 201 (1999), 201-247. Zbl0948.32027 MR1669413 · Zbl 0948.32027 · doi:10.1007/s002200050554 |
| [6] | J. Chen and J. Li, Quaternionic maps between hyperkähler manifolds, J. Differential Geom. 55 (2000), 355-384. Zbl1067.53035 MR1847314 · Zbl 1067.53035 |
| [7] | J. Chen and J. Li, Mean curvature flow of surface in \(4\)-manifolds, Adv. Math. 163 (2001), 287-309. Zbl1002.53046 MR1864836 · Zbl 1002.53046 · doi:10.1006/aima.2001.2008 |
| [8] | J. Chen and G. Tian, Minimal surfaces in Riemannian 4-manifolds, Geom. Funct. Anal. 7 (1997), 873-916. Zbl0891.53042 MR1475549 · Zbl 0891.53042 · doi:10.1007/s000390050029 |
| [9] | S. S. Chern and J. Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), 59-83. Zbl0521.53050 MR692106 · Zbl 0521.53050 · doi:10.2307/2374381 |
| [10] | S. Donaldson and R. Thomas, Gauge theory in higher dimensions, In: “The Geometric Universe: Science, Geometry and the work of Roger Penrose”, S. A. Huggett et al. (eds), Oxford Univ. Press, 1998, pp. 31-47. Zbl0926.58003 MR1634503 · Zbl 0926.58003 |
| [11] | L. C. Evens, Partial regularity for stationary harmonic maps, Arch. Rat. Mech. Anal. 116 (1991), 101-112. Zbl0754.58007 · Zbl 0754.58007 · doi:10.1007/BF00375587 |
| [12] | J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 589-640. Zbl0627.58019 MR848842 · Zbl 0627.58019 |
| [13] | J. M. Figuroa-O’Farrill, C. Köhl and B. Spence, Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves, Nucl. Phys. B521 (1998), 419-443. Zbl0954.53019 MR1635760 · Zbl 0954.53019 · doi:10.1016/S0550-3213(98)00285-5 |
| [14] | D. Joyce, Hypercomplex algebraic geometry, Quart. J. Math. 49 (1998), 129-162. Zbl0924.14002 MR1634730 · Zbl 0924.14002 · doi:10.1093/qjmath/49.194.129 |
| [15] | F. H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Ann. Math. 149 (1999), 785-829. Zbl0949.58017 MR1709303 · Zbl 0949.58017 · doi:10.2307/121073 |
| [16] | J. Li and G. Tian, A blow-up formula for stationary harmonic maps, Internat. Math. Res. Notices 14 (1998), 735-755. Zbl0944.58010 MR1637101 · Zbl 0944.58010 · doi:10.1155/S1073792898000440 |
| [17] | C. Mamone and S. Salamon, Yang-Mills fields on quaternionic spaces. Nonlinearity 1 (1988), 517-530. Zbl0681.53037 MR967469 · Zbl 0681.53037 · doi:10.1088/0951-7715/1/4/002 |
| [18] | Y. Nagatomo and T. Nitta, \(k\)-instantons on \(G_2(C^{n+2})\) and stable vector bundles. Math. Z. 232 (1999), 721-737. Zbl0945.32009 MR1727550 · Zbl 0945.32009 · doi:10.1007/PL00004780 |
| [19] | Y. S. Poon and K. Galicki, Duality and Yang-Mills fields on quaternionic Kähler manifolds. J. Math. Phys. 32 (1991), 1263-1268. Zbl0729.53033 MR1103479 · Zbl 0729.53033 · doi:10.1063/1.529501 |
| [20] | J. Rawnsley, f-structures, f-twistor spaces and harmonic maps, In: “Geometry Seminar ‘Luigi Bianchi’, II - 1984”, E. Vesentini (ed.), Lect. Nothes Math. 1164, Springer, Berlin, 1985, 85-159. Zbl0592.58009 MR829229 · Zbl 0592.58009 |
| [21] | S. Salamon, Harmonic and holomorphic maps, In: “Geometry Seminar ‘Luigi Bianchi’, II - 1984”, E. Vesentini (ed.), Lect. Notes Math. 1164, Springer, Berlin, 1985, 161-224. Zbl0591.53031 MR829230 · Zbl 0591.53031 |
| [22] | R. Schoen, Analytic aspects of harmonic maps, In: “Seminar on nonlinear Partial Differential equations”, S. S. Chern (ed.), M.S.R.I. Publications 2, Springer-Verlag, New-York, 1984, 321-358. Zbl0551.58011 MR765241 · Zbl 0551.58011 |
| [23] | L. Simon, Rectifiability of the singular set of energy minimizing maps, Calc. Var. Partial Differential Equations 3 (1995), 1-65. Zbl0818.49023 MR1384836 · Zbl 0818.49023 · doi:10.1007/BF01190891 |
| [24] | A. Swann, Quaternionic Kähler Geometry and the Fundamental 4-form, In: “Proc. Curvature Geom. workshop”, C. T. J. Dodson (ed.), ULDM Publications Lancaster, 1989, 165-173. Zbl0744.53020 MR1089891 · Zbl 0744.53020 |
| [25] | J. Sacks and K. Uhlenbeck, The existence of minimal immersions of \(2\)-spheres, Ann. of Math. (2) 113 (1981), 1-24. Zbl0462.58014 MR604040 · Zbl 0462.58014 · doi:10.2307/1971131 |
| [26] | D. Widdows, A Dolbeault-type double complex on quaternionic manifolds, Asian J. Math. 6 (2002), 253-275. Zbl1029.58015 MR1928630 · Zbl 1029.58015 |
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.
