Cohomogeneity one Kähler-Ricci solitons under a Heisenberg group action and related metrics

Abstract

We show that integrability of an almost complex structure in complex dimension $m$ is equivalent, in the presence of an almost hermitian metric, to $m(m-1)$ equations involving what we call shear operators. Inspired by this, we give an ansatz for Kähler metrics in dimension $m>1$, for which at most $m-1$ of these shear equations are non-trivial. The equations for gradient Kähler-Ricci solitons in this ansatz are frame dependent PDEs, which specialize to ODEs under extra assumptions. Metrics solving the latter system include a restricted class of cohomogeneity one metrics, and we find among them complete expanding gradient Kähler-Ricci solitons under the action of the $(2m-1)$-dimensional Heisenberg group, and some incomplete steady solitons. We examine curvature properties and asymptotics for the former Ricci solitons. In another special case of the ansatz we present, for $m=2$, a class of complete metrics of a more general type which we call gradient Kähler-Ricci skew-solitons, which are cohomogeneity one under the Euclidean plane group action. This paper continues research started in [MR, AM2].

In this version we correct a few statements related to curvature and distance, and add material on asymptotics (section 5.5) and on curvature bounds. arXiv admin note: text overlap with arXiv:2007.06471