Type I ancient compact solutions of the Yamabe flow

TL;DR

This paper constructs new rotationally symmetric Type I ancient compact solutions to the Yamabe flow that converge to self-similar solutions moving in opposite directions as time approaches negative infinity.

Contribution

It introduces a novel class of ancient compact solutions to the Yamabe flow with specific symmetry and asymptotic properties, expanding understanding of the flow's solution space.

Findings

Existence of new ancient compact solutions

Solutions converge to self-similar solutions as t → -∞

Solutions are rotationally symmetric and Type I

Abstract

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.