Higher-dimensional flying wing Steady Ricci Solitons

For any $n\geq 4$, we construct an $(n-2)$-parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at a critical point of the soliton potential. Among them lies an $(n-3)$-parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author and produced new examples of steady gradient Ricci solitons with non-negative curvature operator for $n\geq 4$. Our approach is based on constructing continuous families of Ricci flows smoothing emanating from continuous families of spherical polyhedra which still preserves symmetry. This is built upon a new stability result of Ricci flows with scaling invariant estimates. As another application of the method, we prove the stability of asymptotically conical expanding solitons constructed by Deruelle under $L^\infty$ perturbation of links. In particular, the $C^0$-convergence of smooth links implies the smooth convergence of the expanding solitons.

💡 Research Summary

The paper by Chan, Lai, and Lee presents a major advance in the construction of steady gradient Ricci solitons in dimensions $n\ge4$ with non‑negative curvature operator. Building on the “flying wing” solitons introduced by the second author, the authors produce an $(n-2)$‑parameter family of steady solitons whose Ricci tensor eigenvalues at the unique critical point of the potential function can be prescribed arbitrarily (subject only to the natural normalization $\lambda_1\le\cdots\le\lambda_{n-1}$ and $\lambda_n=\lambda_{n-1}$). Moreover, when the eigenvalues are constant on prescribed blocks, the resulting solitons enjoy higher symmetry $O(i_2-i_1)\times\cdots\times O(i_{k+1}-i_k)$ and form an $(n-3)$‑parameter subfamily of non‑collapsed examples.

The construction proceeds in two conceptual stages. First, the authors build a continuous family of smooth metrics $g_{i,x}$ on the sphere $S^{n-1}$, parametrized by a simplex $\Omega$ of dimension $n-2$, such that each metric has $Rm\ge1$ and its volume collapses to zero as $i\to\infty$. By the uniqueness of expanding solitons with $Rm\ge0$, each $g_{i,x}$ lifts to an expanding Ricci soliton $E_{i,x}$ on $\mathbb R^n$ that is asymptotic to the cone over $(S^{n-1},g_{i,x})$, has scalar curvature $R=1$ at its tip, and retains $Rm\ge0$. As $i\to\infty$, the expanding solitons converge to steady solitons; the vertices of $\Omega$ correspond to the known models (Bryant soliton, $R\times$Bryant, …, $R^{n-2}\times$ Cigar). Mapping each expanding soliton to the list of Ricci eigenvalues at its tip yields a smooth map $\Omega\to\Delta$, where $\Delta$ is an $(n-2)$‑simplex of admissible eigenvalue tuples. Surjectivity of this map is proved by a degree‑theoretic argument, giving a smooth family ${S_x}_{x\in\Delta}$ of steady solitons with the prescribed eigenvalues.

The second, technically more demanding stage deals with the fact that the initial metrics on $S^{n-1}$ are not smooth but have polyhedral singularities arising from iterated spherical suspensions. To run Ricci flow from such data, the authors develop a novel weak‑stability theory for Ricci‑DeTurck flow. Theorem 1.3 states that if a background Ricci flow $\tilde g(t)$ satisfies scaling‑invariant bounds $|Rm|\le\alpha t^{-1}$, injectivity radius $\ge\sqrt\alpha,t^{1/2}$, and $Rm\ge-1$, then any initial metric $g_0$ that is $L^\infty$‑close to $\tilde g(0)$ admits a Ricci‑DeTurck solution on $