Special solutions to the Type IIA flow

TL;DR

This paper studies special solutions to the source-free Type IIA flow on symplectic half-flat SU(3)-structures, demonstrating existence of ancient, immortal, and eternal solutions, especially on homogeneous spaces.

Contribution

It establishes the existence of various special solutions to the Type IIA flow under symplectic half-flat conditions, including self-similar solutions with Hermitian Ricci tensor.

Findings

Existence of ancient, immortal, and eternal solutions.

Self-similar solutions with Hermitian Ricci tensor.

Applicability to homogeneous spaces with invariant structures.

Abstract

We consider the source-free Type IIA flow introduced by Fei-Phong-Picard-Zhang, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal solutions to the flow, provided that the initial symplectic half-flat structure satisfies suitable properties. In particular, we prove that the solution starting at a symplectic half-flat structure with Hermitian Ricci tensor is ancient and evolves self-similarly by scaling the initial datum. These results apply to all known (locally) homogeneous spaces admitting invariant symplectic half-flat SU(3)-structures.