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Infinitely many non-collapsed steady Ricci solitons on complex line bundles

Hanci Chi

Abstract

We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles $O(k)$ over $\mathbb{CP}^{2m+1}$, where the base space is not necessarily Kähler--Einstein. Each $O(k)$ with $k\in [3,2m+1]$ admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each $O(k)$ with $k\geq 3$, the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

Infinitely many non-collapsed steady Ricci solitons on complex line bundles

Abstract

We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles over , where the base space is not necessarily Kähler--Einstein. Each with admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each with , the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

Paper Structure

This paper contains 9 sections, 22 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. Cohomogeneity one equation
  3. Local Analysis
  4. Global analysis
  5. $k\leq 2$
  6. $k\geq 3$
  7. Asymptotics
  8. Non-Einstein Ricci solitons
  9. Einstein metrics

Key Result

Theorem 1.1

For $k\in \{1,2\}$, there exists a continuous $3$-parameter family of complete Ricci solitons $\{\xi(k,\theta,s_4,s_5)\mid \theta\in (0,\pi),s_4\geq 0, s_5\geq 0\}$ on $\mathcal{O}(k)$. In particular,

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 3.1
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 33 more