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Abstract: We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts. [Copyright &y& Elsevier]

We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]