Energy identities and monotonicity for evolving $$\varvec{k}$$ k -forms on moving Riemannian spaces

Abstract

In this note, we establish energy identities for vector bundle-valued k-forms satisfying suitable heat-type equations, where the ambient space is equipped with a simultaneously evolving metric tensor. We show that if this metric is suitably controlled or a gradient shrinking Ricci soliton, then this energy identity reduces to a monotonicity formula. As special cases, we obtain new monotonicity formulae for the harmonic map and Yang–Mills heat flows in this more general setting, unifying and generalising results due to Struwe, Chen and Struwe, Hamilton, and Chen and Shen in the process.