A family of 3d steady gradient solitons that are flying wings

TL;DR

This paper constructs and analyzes specific families of 3D and higher-dimensional steady gradient Ricci solitons with unique geometric properties, confirming conjectures and expanding understanding of their structure.

Contribution

It introduces new examples of steady gradient Ricci solitons in 3D and higher dimensions, verifying a conjecture and exploring their curvature and collapse properties.

Findings

3D flying wing solitons verify Hamilton's conjecture

3D flying wings are collapsed with non-vanishing scalar curvature at infinity

Higher-dimensional solitons are non-collapsed with positive curvature operator

Abstract

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n\ge 4$, we find a family of $\mathbb{Z}_2\times O(n-1)$-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.