On Ricci Solitons with Isoparametric Potential Functions

This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors.

💡 Research Summary

The paper investigates complete gradient Ricci solitons (GRS) whose potential function is isoparametric, i.e., transnormal with constant mean curvature on each level set. After recalling basic definitions, the authors prove two main structural results.

In the steady case (λ = 0) they show that any non‑trivial complete GRS with a transnormal potential must possess a singular level set of codimension greater than one, which corresponds to the global maximum of the potential. The proof proceeds by contradiction: assuming ∇f never vanishes, the manifold would be diffeomorphic to a product P × ℝ, and the transnormality forces the shape operator L to satisfy a Riccati‑type system. An analysis of the resulting ODEs shows that either the scalar curvature vanishes (forcing the soliton to be trivial) or the second derivative of f cannot maintain the required sign, leading to a contradiction. Hence a critical point of f exists, and the associated level set cannot be a regular hypersurface; it must be a singular foil.

For shrinking solitons (λ > 0) the authors use known quadratic growth estimates for the potential to guarantee compactness of all level sets. If no singular foil exists, every critical level set would be a hypersurface, implying the manifold is diffeomorphic to P × ℝ or P × S¹. The latter is ruled out because the quantity f′ − tr L would be periodic while its derivative is strictly positive (λ + tr L² ≥ λ), an impossibility. Consequently, a shrinking soliton either has a singular foil or splits globally as ℝ × P. In the compact shrinking case, the existence of both a global minimum and maximum forces exactly two singular foils.

The paper then studies a specific ansatz: a line or circle bundle over a Kähler‑Einstein base (N^{n‑1}(k), Rc_N = k Id) with metric
g = dt² + H(t)² η⊗η + F(t)² π* g_N,
where η is the connection 1‑form satisfying dη = q π* ω_N (q = 0 or 1). For steady solitons with q = 1 and potential depending only on t, they prove: (1) H(0)=0 and H is strictly increasing; (2) F is either monotone increasing or attains a global minimum, and in any case F(t) → ∞ as t → ∞. These properties mirror known behavior of Kähler steady solitons constructed via cohomogeneity‑one methods, and the ansatz includes the Hopf fibration case (N = ℂP^{n‑1}, q = 1).

Overall, the work demonstrates that the isoparametric condition on the potential function imposes strong topological restrictions on gradient Ricci solitons. In the steady case a singular foil is unavoidable; in the shrinking case its presence or a global product structure exhausts the possibilities. The detailed analysis of the ansatz provides concrete asymptotic information for a broad class of examples, linking the abstract theory to explicit constructions in both Kähler and non‑Kähler settings.