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Quaternionic maps between hyper-Kähler manifolds. (English) Zbl 1067.53035
Recall that the quaternionic analog of a Kähler manifold (i.e. a Riemannian manifold with a “compatible” complex structure on its tangent bundle) is a hyper-Kähler manifold which is, by definition, a Riemannian manifold \(M\) with three covariantly constant orthogonal automorphisms \(I\), \(J\), and \(K\) of the tangent bundle \(TM\), satisfying \(I^{2}=J^{2}=K^{2} =IJK=-\text{id}\), with \(I,J,K\) implementing the scalar multiplications by the standard generators \(i,j,k\) of the quaternion division ring on the tangent spaces. Since for any \(A\in SO( 3) \) acting on \(\mathbb{R}^{3}\) spanned by \(\{ i,j,k\} \), \(( Ai,Aj,Ak) \) forms another triple of “standard” generators of the quaternion division ring, a smooth map \(u:M\to N\) between hyper-Kähler manifolds that respects the quaternionic structures and is called a quaternionic map, is characterized by the defining property that \(\sum_{\alpha\beta}A_{\alpha\beta}\mathfrak{I}^{\beta}\circ du\circ J^{\alpha}=du\) for some \(A\in SO( 3) \), where \(du:TM\to TN\) is the total derivative and \(\{ J^{\alpha}\} _{1\leq\alpha\leq3},\{ \mathfrak{I}^{\beta}\} _{1\leq\beta\leq3}\) are the automorphism triples defining the quaternionic structures on \(M,N\), respectively. Note that to any hyper-Kähler manifold \(M\), there is associated an \(\mathbb{S}^{2}\)-family, called the hyper-Kähler \(\mathbb{S}^{2}\) of \(M\), of complex structures on \(TM\), namely, the family of complex structures defined by \(aI+bJ+cK\) for all \(( a,b,c) \in\mathbb{S}^{2}\).
Like a holomorphic map between Kähler manifolds, a quaternionic map \(u:M\to N\) between hyper-Kähler manifolds also minimizes the energy functional \(E( u) :=\frac{1}{2}\int_{M}g^{ij}h_{mn} \partial_{i}u^{m}\partial_{j} u^{n}\,dV\) in its homotopy class and hence is harmonic, where \(g,h\) are the Riemannian metrics on \(M,N\), respectively. However the authors show that it is not true that every quaternionic map \(u:M\to N\) is a holomorphic map with respect to some suitable complex structures on \(M,N\). In fact, they give a necessary and sufficient condition for a quaternionic map \(u:M\to N\) to be a holomorphic map with respect to some complex structures in the hyper-Kähler \(\mathbb{S}^{2}\) of \(M\) and \(N\), and provide a concrete example violating this condition. The authors also analyze the structure of the blow-up set of a sequence of quaternionic maps \(u_{k}\) from a hyper-Kähler surface (i.e. a hyper-Kähler manifold of real dimension four) with uniformly bounded energy \(E( u_{k}) \), of which there is always a subsequence converging weakly to a stationary quaternionic map. Furthermore, it is shown that for a stationary quaternionic map \(u:M\to N\) from a hyper-Kähler surface \(M\) to a hyper-Kähler manifold \(N\) with its hyper-Kähler structure defined by a real analytic metric, the singular set of \(u\) is of Hausdorff dimension at most one and is rectifiable if the dimension is one.

MSC:
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
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