Levy Laplacian on manifold and heat flows of differential forms

TL;DR

This paper explores the Levy Laplacian on manifolds, establishing its various definitions, analyzing the heat equation associated with it, and studying the long-term behavior of solutions using heat flows of differential forms.

Contribution

It proves the equivalence of different definitions of the Levy Laplacian on path manifolds and investigates the asymptotic behavior of heat equation solutions involving differential forms.

Findings

Equivalence of Levy Laplacian definitions on path manifolds

Solutions tend to locally constant functionals over time

Heat flows of differential forms are used to construct solutions

Abstract

The Levy Laplacian is an infinite-dimensional differential operator, which is interesting for its connection with the Yang-Mills gauge fields. The article proves the equivalence of various definitions of the Levy Laplacian on the manifold of $H^1$-paths on a Riemannian manifold. The heat equation with the Levy Laplacian is considered. The tendency of some solutions of this heat equation to the locally constant functionals as time tends to infinity is studied. These solutions are constructed using heat flows of differential forms on the compact Riemannian manifold.