An ancient Ricci flow emerging from Taub-Bolt

TL;DR

The paper proves the existence of a nontrivial ancient Ricci-flow solution emerging from a Taub-Bolt metric on $\mathbb{CP}^2\setminus\{\mathrm{pt}\}$. It establishes linear instability of Taub-Bolt by constructing an unstable eigenmode of the Lichnerowicz Laplacian via a variational functional $\mathbf{a}$ and a gauge-fixed perturbation $h$ that decays exponentially in the radial coordinate. The main construction uses Ricci-DeTurck flow with an unstable mode, proving the existence of ancient solutions that converge to the Taub-Bolt metric (modulo diffeomorphisms) as $t\to -\infty$, and then pulls back to a Ricci-flow solution. Regularity and exponential decay of the eigenmode are shown, and a careful variational, frequency-analytic, and parabolic-estimate framework yields the necessary $C^m$ and $H^m$ bounds to obtain a nontrivial ancient solution in the non-compact ALF setting. This advances understanding of stability phenomena for non-compact Ricci-flat geometries and provides a concrete ancient solution emerging from an explicit ALF metric.

Abstract

This paper proves that there exists a non-trivial ancient solution to the Ricci flow emerging from the Taub-Bolt metric.

Theorems & Definitions (39)

• Theorem 1.1
• Definition 1.2
• Definition 1.3: Ses1 Definition 2
• Theorem 1.4
• Theorem 1.5: Has1, Theorem 1.2
• Theorem 1.6
• Definition 3.1
• Theorem 3.2
• proof
• proof