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Higher-dimensional flying wing Steady Ricci Solitons

Pak-Yeung Chan, Yi Lai, Man-Chun Lee

TL;DR

We address the problem of constructing higher-dimensional steady gradient Ricci solitons with non-negative curvature operator and prescribed Ricci eigenvalues at the unique critical point, extending the flying wings program to $n\ge 4$. The authors develop a framework for evolving spaces with isolated or polyhedral singularities by gluing expanding solitons and obtaining Ricci flows from non-smooth data, controlled via weak stability of Ricci-DeTurck flow and scaling-invariant estimates. They establish a smoothing mechanism for spherical polyhedra, preserving symmetry and enabling continuous dependence on the link data, and then realize steady solitons as limits of expanding solitons associated to vertex links, parameterized by a simplex in eigenvalue space. The main contributions include the existence of an $(n-2)$-parameter family of steady solitons with prescribed Ricci eigenvalues, an $(n-3)$-parameter non-collapsed subfamily, and a robust stability/smoothing theory connecting singular initial data to smooth solitons. Together, these results provide a broad, structurally rich class of higher-dimensional flying wings with non-negative curvature, linking singular space smoothing to soliton construction and bridging Ricci flow from non-smooth data with soliton theory.

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 the unique 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. Our approach is based on constructing continuous families of smooth Ricci flows emanating from continuous families of spherical polyhedra. This is built upon a combination of a new stability result of Ricci flows with scaling invariant estimates and the method of Gianniotis-Schulze in regularizing manifolds with singularities.

Higher-dimensional flying wing Steady Ricci Solitons

TL;DR

We address the problem of constructing higher-dimensional steady gradient Ricci solitons with non-negative curvature operator and prescribed Ricci eigenvalues at the unique critical point, extending the flying wings program to . The authors develop a framework for evolving spaces with isolated or polyhedral singularities by gluing expanding solitons and obtaining Ricci flows from non-smooth data, controlled via weak stability of Ricci-DeTurck flow and scaling-invariant estimates. They establish a smoothing mechanism for spherical polyhedra, preserving symmetry and enabling continuous dependence on the link data, and then realize steady solitons as limits of expanding solitons associated to vertex links, parameterized by a simplex in eigenvalue space. The main contributions include the existence of an -parameter family of steady solitons with prescribed Ricci eigenvalues, an -parameter non-collapsed subfamily, and a robust stability/smoothing theory connecting singular initial data to smooth solitons. Together, these results provide a broad, structurally rich class of higher-dimensional flying wings with non-negative curvature, linking singular space smoothing to soliton construction and bridging Ricci flow from non-smooth data with soliton theory.

Abstract

For any , we construct an -parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at the unique critical point of the soliton potential. Among them lies an -parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author. Our approach is based on constructing continuous families of smooth Ricci flows emanating from continuous families of spherical polyhedra. This is built upon a combination of a new stability result of Ricci flows with scaling invariant estimates and the method of Gianniotis-Schulze in regularizing manifolds with singularities.
Paper Structure (20 sections, 23 theorems, 189 equations, 2 figures)

This paper contains 20 sections, 23 theorems, 189 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Flying wing steady solitons
  3. Ricci flow starting from non-smooth initial data
  4. Preliminaries: steady Ricci solitons
  5. Ricci flow from spaces with isolated singularities
  6. Expander with $\mathrm{Rm}\geq 0$
  7. Existence of Ricci flow
  8. Stability of Ricci flows
  9. Relation between Ricci flow and Ricci-DeTurck flow
  10. Main result: weak stability of Ricci-DeTurck flow
  11. Evolution equation
  12. Some a-priori estimates
  13. Heat kernel estimates
  14. Proof of weak stability
  15. Smoothing singular metrics on spheres and stability
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $n\ge4$, for any $0\le\lambda_1\le\cdots\le\lambda_{n-1}$ with $\lambda_1+\cdots+\lambda_{n-2}+2\lambda_{n-1}=1$, there exists an $n$-dimensional steady gradient soliton $(M,g,f,p)$ with $\mathrm{Rm}\geq 0$, $R(p)=1$, and the eigenvalues of Ricci curvature at $p$ are equal to $(\lambda_1,\cdots,

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 52 more