A heat flow for special metrics

TL;DR

This paper investigates a geometric flow on seven-manifolds driven by a functional related to $G_2$-holonomy, proving existence, uniqueness, and convergence results for the flow near critical points.

Contribution

It introduces a natural functional on positive 3-forms, analyzes its negative gradient flow, and establishes long-term existence and convergence near critical points.

Findings

Short-time existence and uniqueness of the flow

Global existence for initial conditions close to critical points

Flow converges to a critical point modulo diffeomorphisms

Abstract

On the space of positive 3-forms on a seven-manifold, we study a natural functional whose critical points induce metrics with holonomy contained in $G_2$. We prove short-time existence and uniqueness for its negative gradient flow. Furthermore, we show that the flow exists for all times and converges modulo diffeomorphisms to some critical point for any initial condition sufficiently $C^\infty$-close to a critical point.