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

Hanci Chi

Abstract

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.

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

Abstract

We establish the existence of two 3-parameter families of non-Einstein, non-shrinking Ricci solitons: one on and one on . 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 , which contains a 1-parameter subfamily of asymptotically paraboloidal steady Ricci solitons based on the Bourguignon--Karcher sphere.

Paper Structure

This paper contains 9 sections, 27 theorems, 109 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Cohomogeneity one Ricci soliton equation
  3. Existence of complete Ricci solitons
  4. Asymptotics
  5. Steady Ricci-solitons
  6. Expanding Ricci solitons
  7. Ricci solitons on $\mathbb{O}^2$
  8. Solitons with positive sectional curvatures
  9. Visual Summary

Key Result

Theorem 1.3

There exists a continuous 3-parameter family of complete $Sp(m+1)U(1)$-invariant Ricci solitons $\{\zeta(s_1,s_2,s_3,s_4)\mid (s_1,s_2,s_3,s_4)\in \mathbb{S}^3, s_1,s_4>0,s_2,s_3\geq 0\}$ on $\mathbb{HP}^{m+1}\backslash\{*\}$.

Figures (2)

  • Figure 1: Ricci solitons on $\mathbb{HP}^{m+1}\backslash\{*\}$
  • Figure 2: Ricci solitons on $\mathbb{H}^{m+1}$

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 2.1
  • proof
  • ...and 45 more