Kähler Solitons, Contact Structures, and Isoparametric Functions
Hung Tran
TL;DR
The paper classifies complete gradient Kähler-Ricci solitons in real dimension four under the hypothesis that the soliton potential $f$ and scalar curvature $ m S$ are functionally dependent, showing that such GKRS are necessarily cohomogeneity-one and intimately linked to isoparametric functions and deformed Sasakian geometry. The authors develop a Calabi-type, first-order framework for the soliton equations on level sets of $f$, reveal a deformed contact/Sasakian structure on regular level sets, and establish a Sasakian-model/directed-submersion description over Kähler-Einstein bases. In dimension four, the Hessian of $f$ has two constant eigenvalues, forcing either a deformed Sasakian fibration over a constant-curvature base or a local product, thereby recovering all known $U(2)$-invariant models and providing explicit ODE reductions for the metric components. The work bridges GKRS with contact geometry and isoparametric theory, yielding a complete classification in the dependent case and connecting it to Calabi-type constructions and Sasakian space-forms. These findings advance the understanding of symmetry, geometric structure, and explicit metrics in GKRS, with potential implications for the broader study of Ricci solitons and complex-geometry flows.
Abstract
All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\SS$. In this article, we consider the case that $f$ and $\SS$ are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.