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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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