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Integrable systems in Riemannian geometry. (English) Zbl 0939.37039

Terng, Chuu Lian (ed.) et al., Surveys in differential geometry. Vol. IV. A supplement to the Journal of Differential Geometry. Integral systems (integrable systems). Lectures on geometry and topology. Cambridge, MA: International Press. 21-81 (1998).
The author surveys several problems in differential geometry which involve equations usually considered as “completely integrable,” including:
(i) the problem of constructing immersed constant mean curvature 2-tori in \(\mathbb{R}^3\), \(\mathbb{S}^3\), or other manifolds, viewed from different perspectives;
(ii) the construction of SU(2)-invariant Einstein metrics from a special case of the Painlevé VI equation, in which a solution in terms of theta functions is available [N. J. Hitchin, J. Differ. Geom. 42, 30-112 (1995; Zbl 0861.53049)];
(iii) the relation between Nahm’s equations and the self-dual Yang-Mills equations (for more on the relation between self-dual Yang-Mills equations and integrable PDEs, see M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Univ. Press (1991; Zbl 0762.35001));
(iv) new results on the explicit construction of hyperkähler metrics via the solution of Nahm’s equations.
For the entire collection see [Zbl 0918.00013].

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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