Higher-dimensional flying wing Steady Ricci Solitons

TL;DR

This paper constructs new higher-dimensional steady gradient Ricci solitons with non-negative curvature operator, expanding the known examples and introducing a novel approach based on Ricci flow smoothing from spherical polyhedra.

Contribution

It introduces a method to generate continuous families of Ricci solitons using Ricci flow smoothing from spherical polyhedra, including new non-collapsed examples in higher dimensions.

Findings

Constructed an $(n-2)$-parameter family of steady gradient Ricci solitons for $n extgreater=4$.

Identified a subfamily of non-collapsed solitons with prescribed eigenvalues.

Proved stability of asymptotically conical expanding solitons under $L^ abla$ perturbations.

Abstract

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.