Laplacian coflows of $G_2$-structures on contact Calabi--Yau 7-manifolds

TL;DR

This paper studies three variants of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi--Yau 3-folds, linking their behavior to modified K"ahler--Ricci flows through dimensional reduction.

Contribution

It introduces new formulations of the Laplacian coflow on contact Calabi--Yau 7-manifolds and connects them to known K"ahler--Ricci flows via dimensional reduction techniques.

Findings

Derived natural modifications of K"ahler--Ricci flow from Laplacian coflows.

Analyzed the behavior of coflows on trivial products and contact Calabi--Yau manifolds.

Established links between $G_2$-structure flows and K"ahler geometry.

Abstract

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi--Yau 3-folds, interpreting their dimensional reductions to the K\"ahler geometry of the base. Precisely, we reduce Ans\"atze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products $CY^3\times S^1$ and on contact Calabi--Yau 7-manifolds, obtaining in each case a natural modification of the K\"ahler--Ricci flow.