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The space of hyperkähler metrics on a 4-manifold with boundary. (English) Zbl 1375.53061

The paper under review deals with the study of hyper-Kähler metrics on a compact 4-manifold \(X\) with boundary \(\partial X=Y\). Denote by \((J_1, J_2, J_3)\) the triple of orthogonal complex structures and by \(\omega=(\omega^1,\omega^2,\omega^3)\) the corresponding triple of 2-forms on \(X\).
This paper is inspired in part by a local thickening result of R. L. Bryant [in: The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press. 346–367 (2010; Zbl 1221.53084)].
It is proved that the moduli space of hyper-Kähler triples \(\omega\) on \(X\), modulo diffeomorphisms which are the identity on the boundary, is a smooth infinite-dimensional manifold and the tangent space is described in terms of triples of closed anti-self-dual 2-forms.
Next, boundary value problems for hyper-Kähler metrics are investigated.
The authors show that the moduli space \({\mathcal M}_X\) of hyper-Kähler triples on X inducing positive mean curvature on the boundary \(Y\), modulo the action of \({\mathcal G}_0\), is a Fréchet manifold and determine its tangent space, where \({\mathcal G}_0\) is the gauge group of orientation-preserving diffeomorphisms of \(X\) which are the identity on \(\partial X\).
Finally, explicit examples coming from gravitational instantons with isometric actions of \(\mathrm{SU}(2)\) are studied.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
58J32 Boundary value problems on manifolds

Citations:

Zbl 1221.53084
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References:

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