|
SIGMA 9 (2013), 075, 21 pages arXiv:1306.3195
https://doi.org/10.3842/SIGMA.2013.075
Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
Mikhail B. Sheftel a and Andrei A. Malykh b
a) Department of Physics, Boğaziçi University 34342 Bebek, Istanbul, Turkey
b) Department of Numerical Modelling, Russian State Hydrometeorlogical University, 98 Malookhtinsky Ave., 195196 St. Petersburg, Russia
Received June 14, 2013, in final form November 19, 2013; Published online November 27, 2013
Abstract
We demonstrate how a combination of our recently developed methods of partner symmetries,
symmetry reduction in group parameters and a new version of the group foliation method can produce
noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant
solutions of CMA satisfying Boyer-Finley equation to non-invariant ones.
Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat
anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with
Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices
of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in
a bounded domain.
Key words:
Monge-Ampère equation; Boyer-Finley equation; partner symmetries; symmetry reduction; non-invariant solutions; group foliation; anti-self-dual gravity; Ricci-flat metric.
pdf (439 kb)
tex (30 kb)
References
- Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional
Riemannian geometry, Proc. Roy. Soc. London Ser. A 362
(1978), 425-461.
- Boyer C.P., Finley III J.D., Killing vectors in self-dual, Euclidean
Einstein spaces, J. Math. Phys. 23 (1982), 1126-1130.
- Calderbank D.M.J., Tod P., Einstein metrics, hypercomplex structures and the
Toda field equation, Differential Geom. Appl. 14 (2001),
199-208, math.DG/9911121.
- Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts
in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
- Dunajski M., Gutowski J., Sabra W., Enhanced Euclidean supersymmetry, 11D
supergravity and SU(∞) Toda equation, J. High Energy
Phys. 2013 (2013), no. 10, 089, 20 pages, arXiv:1301.1896.
- Dunajski M., Mason L.J., Twistor theory of hyper-Kähler metrics with hidden
symmetries, J. Math. Phys. 44 (2003), 3430-3454,
math.DG/0301171.
- Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and
differential geometry, Phys. Rep. 66 (1980), 213-393.
- Hitchin N., Compact four-dimensional Einstein manifolds,
J. Differential Geometry 9 (1974), 435-441.
- Lie S., Über Differentialinvarianten, Math. Ann. 24
(1884), 537-578.
- Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries of the complex
Monge-Ampère equation yield hyper-Kähler metrics without
continuous symmetries, J. Phys. A: Math. Gen. 36 (2003),
10023-10037, math-ph/0305037.
- Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries and non-invariant
solutions of four-dimensional heavenly equations, J. Phys. A: Math.
Gen. 37 (2004), 7527-7545, math-ph/0403020.
- Malykh A.A., Nutku Y., Sheftel M.B., Lift of noninvariant solutions of heavenly
equations from three to four dimensions and new ultra-hyperbolic metrics,
J. Phys. A: Math. Theor. 40 (2007), 9371-9386,
arXiv:0704.3335.
- Malykh A.A., Sheftel M.B., Recursions of symmetry orbits and reduction without
reduction, SIGMA 7 (2011), 043, 13 pages,
arXiv:1005.0153.
- Martina L., Sheftel M.B., Winternitz P., Group foliation and non-invariant
solutions of the heavenly equation, J. Phys. A: Math. Gen.
34 (2001), 9243-9263, math-ph/0108004.
- Mason L.J., Newman E.T., A connection between the Einstein and
Yang-Mills equations, Comm. Math. Phys. 121 (1989),
659-668.
- Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory,
London Mathematical Society Monographs. New Series, Vol. 15, The
Clarendon Press, Oxford University Press, New York, 1996.
- Nutku Y., Sheftel M.B., A family of heavenly metrics, gr-qc/0105088.
- Nutku Y., Sheftel M.B., Differential invariants and group foliation for the
complex Monge-Ampère equation, J. Phys. A: Math. Gen.
34 (2001), 137-156.
- Olver P.J., Applications of Lie groups to differential equations,
Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York,
1986.
- Ovsiannikov L.V., Group analysis of differential equations, Academic Press
Inc., New York, 1982.
- Plebañski J.F., Some solutions of complex Einstein equations,
J. Math. Phys. 16 (1975), 2395-2402.
- Schrüfer E., EXCALC: A differential geometry package, in REDUCE, User's and Contributed
Packages Manual, Version 3.8, Editor A.C. Hearn, Santa Monica, CA, 2003,
333-343.
- Sheftel M.B., Method of group foliation and non-invariant solutions of partial
differential equations. Example: the heavenly equation, Eur.
Phys. J. B 29 (2002), 203-206.
- Sheftel M.B., Malykh A.A., Lift of invariant to non-invariant solutions of
complex Monge-Ampère equations, J. Nonlinear Math. Phys.
15 (2008), suppl. 3, 385-395, arXiv:0802.1463.
- Sheftel M.B., Malykh A.A., On classification of second-order PDEs possessing
partner symmetries, J. Phys. A: Math. Theor. 42 (2009),
395202, 20 pages, arXiv:0904.2909.
- Tod K.P., Scalar-flat Kähler and hyper-Kähler metrics from
Painlevé-III, Classical Quantum Gravity 12 (1995),
1535-1547, gr-qc/0105088.
- Vessiot E., Sur l'intégration des systèmes différentiels qui admettent
des groupes continus de transformations, Acta Math. 28
(1904), 307-349.
- Yau S.T., Calabi's conjecture and some new results in algebraic geometry,
Proc. Nat. Acad. Sci. USA 74 (1977), 1798-1799.
|
|