Explicit construction of self-dual 4-manifolds. (English) Zbl 0855.57028

Joyce, Dominic D.

Duke Math. J. 77, No. 3, 519-552 (1995).

A self-dual metric or conformal structure on a 4-manifold \(M\) is a Riemannian metric \(g\) or conformal class \([g]\) for which the Weyl conformal curvature tensor \(W\) is self-dual; \(M\) is then called a self-dual 4-manifold. Compact self-dual 4-manifolds have been extensively studied, and it is known that very many compact 4-manifolds do admit families of self-dual metrics, but explicit examples of self-dual metrics that can be written down in coordinates are comparatively few. In this paper we provide a geometrical framework within which it is possible to construct self-dual structures by solving a linear rather than a nonlinear equation, and use it to construct some new explicit examples of compact self-dual 4-manifolds.

MSC:

57R57	Applications of global analysis to structures on manifolds
57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Keywords:

self-dual metric; self-dual conformal structure; 4-manifold; conformal curvature tensor; compact self-dual 4-manifolds

References:

[1]	A. Ashtekar, T. Jacobson, and L. Smolin, A new characterization of half-flat solutions to Einstein’s equation , Comm. Math. Phys. 115 (1988), no. 4, 631-648. · Zbl 0642.53079 · doi:10.1007/BF01224131
[2]	M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry , Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425-461. · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[3]	W. G. Brown, Historical note on a recurrent combinatorial problem , Amer. Math. Monthly 72 (1965), 973-977. JSTOR: · Zbl 0136.21204 · doi:10.2307/2313332
[4]	J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I , J. Differential Geom. 23 (1986), no. 3, 309-346. · Zbl 0606.53028
[5]	G. W. Gibbons and S. W. Hawking, Gravitational multi-instantons , Phys. Lett. B 78 (1978), 430-432.
[6]	P. E. Jones and K. P. Tod, Minitwistor spaces and Einstein-Weyl spaces , Classical Quantum Gravity 2 (1985), no. 4, 565-577. · Zbl 0575.53042 · doi:10.1088/0264-9381/2/4/021
[7]	D. D. Joyce, The hypercomplex quotient and the quaternionic quotient , Math. Ann. 290 (1991), no. 2, 323-340. · Zbl 0723.53043 · doi:10.1007/BF01459248
[8]	D. Joyce, Hypercomplex and quaternionic manifolds, part I , Ph.D. thesis, Oxford, 1992. · Zbl 0735.53050
[9]	C. LeBrun, Explicit self-dual metrics on \(\mathbf C\mathrm P_ 2\#\cdots\#\mathbf C\mathrm P_ 2\) , J. Differential Geom. 34 (1991), no. 1, 223-253. · Zbl 0725.53067
[10]	L. J. Mason and E. T. Newman, A connection between the Einstein and Yang-Mills equations , Comm. Math. Phys. 121 (1989), no. 4, 659-668. · Zbl 0668.53048 · doi:10.1007/BF01218161
[11]	P. Orlik and F. Raymond, Actions of the torus on \(4\)-manifolds. I , Trans. Amer. Math. Soc. 152 (1970), 531-559. JSTOR: · Zbl 0216.20202 · doi:10.2307/1995586
[12]	P. Orlik and F. Raymond, Actions of the torus on \(4\)-manifolds. II , Topology 13 (1974), 89-112. · Zbl 0287.57017 · doi:10.1016/0040-9383(74)90001-9
[13]	Y. Sun Poon, Compact self-dual manifolds with positive scalar curvature , J. Differential Geom. 24 (1986), no. 1, 97-132. · Zbl 0583.53054
[14]	Y. Sun Poon, Algebraic dimension of twistor spaces , Math. Ann. 282 (1988), no. 4, 621-627. · Zbl 0665.32014 · doi:10.1007/BF01462887
[15]	S. M. Salamon, Differential geometry of quaternionic manifolds , Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31-55. · Zbl 0616.53023
[16]	C. H. Taubes, The existence of anti-self-dual conformal structures , J. Differential Geom. 36 (1992), no. 1, 163-253. · Zbl 0822.53006