Kähler Solitons, Contact Structures, and Isoparametric Functions

Abstract All known examples of 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 $${\mathrm S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> . In this article, we consider the case that f and $${\mathrm S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> 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.