Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration

We establish the existence of two 3-parameter families of non-Einstein, non-shrinking Ricci solitons: one on $\mathbb{H}^{m+1}$ and one on $\mathbb{HP}^{m+1}\backslash{*}$. Each family includes a continuous 1-parameter subfamily of asymptotically paraboloidal (non-collapsed) steady Ricci solitons, with the Jensen sphere as the base. Additionally, we extend this result by proving the existence of a 2-parameter family on $\mathbb{O}^2$, which contains a 1-parameter subfamily of asymptotically paraboloidal steady Ricci solitons based on the Bourguignon–Karcher sphere.

💡 Research Summary

The paper investigates non‑shrinking (i.e. expanding or steady) Ricci solitons of cohomogeneity one that arise from the quaternionic Hopf fibration. By exploiting the triple of groups ((K,H,G)=(Sp(m)\Delta U(1),,Sp(m)Sp(1)U(1),,Sp(m+1)U(1))), the isotropy representation of the principal orbit (G/K) splits into three irreducible summands (1\oplus2\oplus4m). This three‑summand structure allows the authors to write the cohomogeneity‑one metric as
(g=dt^{2}+a^{2}(t)Q|{1}+b^{2}(t)Q|{2}+c^{2}(t)Q|_{4m})
and to derive the full Ricci‑soliton system (2.3) for the metric functions (a,b,c) together with the potential (f).

A crucial step is the change of independent variable (d\eta=(\operatorname{tr}L-\dot f)dt) and the introduction of seven new variables ((X_{1,2,3},Y_{1,2,3},W)) that encode the normalized shape operator and metric ratios. In this formulation the system becomes an autonomous ODE (2.12) with a conserved quantity (Q) and an averaged curvature (H). The authors prove that the region
(RS={Q\le0,;H\le1,;W\ge0,;Y_{i}\ge0})
is invariant under the flow (Proposition 2.1). Within (RS) the equalities (Q=0,;H=1) characterize Einstein metrics, (W=0) characterizes steady solitons, and (W>0) characterizes expanding solitons. Additional invariant subsets (RS_{FS}, RS_{KE}, RS_{round}) correspond respectively to circle bundles over (\mathbb{CP}^{2m+1}), Kähler‑Ricci solitons, and round (S^{3})-bundles over (\mathbb{HP}^{m}).

The initial value problem is treated at two types of singular orbits: (i) collapse of the (S^{3}) fibre to the quaternionic projective space (\mathbb{HP}^{m}) (condition (2.8)), and (ii) total collapse to a point (condition (2.9)). Four parameters ((s_{1},s_{2},s_{3},s_{4})) arise from the squashing of the two first summands, the generalized mean curvature of the principal orbit, and the second derivative of the potential at the singular orbit. Scaling of the whole solution corresponds to moving on the unit sphere (S^{3}) (or (S^{2}) in the octonionic case).

The main existence results are:

• Theorem 1.3 – A continuous three‑parameter family ({\zeta(s_{1},s_{2},s_{3},s_{4})}) of complete (Sp(m+1)U(1))-invariant Ricci solitons on (\mathbb{HP}^{m+1}\setminus{*}).
  – When (s_{2}=0) the soliton is steady, non‑Einstein, and asymptotically paraboloidal (AP) with the Jensen sphere (S^{4m+3}) as the base of the paraboloid.
  – When (s_{2}>0) the steady soliton is asymptotically cigar‑paraboloidal (ACP) with a non‑Kähler (\mathbb{CP}^{2m+1}) base.
  – If (s_{3}>0) the metric becomes an expanding, asymptotically conical (AC) soliton.

• Theorem 1.4 – An analogous three‑parameter family ({\gamma(s_{1},s_{2},s_{3},s_{4})}) on the hyperbolic space (\mathbb{H}^{m+1}). The same parameter regimes give AP steady solitons based on the Jensen sphere, ACP steady solitons based on the Fubini–Study (\mathbb{CP}^{2m+1}), and AC expanding solitons.

• Theorem 1.5 – A two‑parameter family ({\tilde\gamma(s_{1},s_{3},s_{4})}) of complete (Spin(9))-invariant solitons on the octonionic plane (\mathbb{O}^{2}).
  – For (s_{4}=0) the metric is Ricci‑flat and AC with the Bourguignon–Karcher sphere (S^{15}) as the cone base.
  – For (s_{4}>0) the steady soliton is non‑Einstein, AP, again with the Bourguignon–Karcher sphere as the paraboloid base.
  – Positive (s_{3}) yields expanding solitons that are either asymptotically hyperbolic (AH) when Einstein, or AC non‑Einstein otherwise.

• Theorem 1.8 – By taking (s_{1}) sufficiently small, one obtains a one‑parameter subfamily of positively curved steady solitons on (\mathbb{H}^{4m+4}) and on (\mathbb{O}^{2}). These solitons are non‑collapsed, have volume growth (t^{n/2}) (with (n) the dimension of the principal orbit), and their asymptotic paraboloids are based on the Jensen sphere or the Bourguignon–Karcher sphere respectively. Because the bases are non‑standard spheres, these solitons are not asymptotically cylindrical in the sense of Brendle; consequently they provide new models for Ricci‑flow singularities beyond the Bryant soliton.

Methodologically, the paper builds on the initial value theory of cohomogeneity‑one Einstein metrics (Eschenburg–Wang 2000) and its extension to Ricci solitons (Buzano 2011). The invariant set (RS) and the monotonicity of (Q) and (H) replace the more delicate shooting arguments used in earlier works on Kähler‑Ricci solitons. The authors also adapt the coordinate change introduced by Dancer–Wang (2009) to handle the extra summand and the non‑shrinking case.

The results significantly broaden the landscape of known non‑shrinking Ricci solitons. Prior examples (Bryant soliton, Kähler‑Ricci solitons on line bundles, Wink solitons on quaternionic Hopf bundles) typically involve one or two summands and yield either cylindrical or conical asymptotics. Here, the three‑summand structure produces families with richer parameter spaces, new asymptotic types (AP and ACP), and examples with positive sectional curvature that are not rotationally symmetric. The appearance of non‑standard sphere bases (Jensen, Bourguignon–Karcher) suggests that the geometry at infinity can be far more flexible than previously thought.

The paper concludes with visual illustrations (Figures 1–2) of the parameter space and asymptotic regimes, and outlines future directions: extending the analysis to the octonionic Hopf fibration in higher dimensions, studying stability and uniqueness of the new steady solitons, and exploring their role as singularity models in Ricci flow.