Hyperkähler metrics of cohomogeneity one (Q677516)

The intractability of the Einstein equations has led mathematicians interested in Einstein metrics to impose simplifying assumptions. One such assumption is that of homogeneity, when the Einstein condition may be expressed purely algebraically. Another natural restriction is to consider metrics of cohomogeneity one, that is, where the generic orbit of the isometry group has real codimension one. For cohomogeneity-one metrics the Einstein condition becomes a system of non-linear ordinary differential equations. Although considerable progress has been made, a classification of complete cohomogeneity-one Einstein metrics still seems far off. The authors prove a classification theorem for cohomogeneity-one metrics which satisfy the stronger condition of being hyperkähler: Let \(M\) be an irreducible hyperkähler manifold of dimension greater than 4 and of cohomogeneity one with respect to a compact simple Lie group \(G\). Then \(M\) is an open subset of the cotangent bundle of complex projective space with the Calabi metric [\textit{E. Calabi}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 269-294 (1979; Zbl 0431.53056)], or the Swann space \(\mathcal U(N)\) [\textit{A. F. Swann}, Math. Ann. 289, 421-450 (1991; 718.53051)] (with its standard metric), where \(N\) is a compact Wolf space. If \(M\) is complete, then \(M\) is \(T^\ast\mathbb C\)P\((n)\) with the Calabi metric.