Ricci flow coupled with harmonic map flow

Abstract

We investigate a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map φ from M to some closed target manifold N , ∂ ∂t

On gradient solitons of the Ricci–Harmonic flow

“…Proof Identity (2.1) can be found implicitly in [10,Chapter 4]. For the sake of completeness, we include a different proof here.…”

“…As in Hamilton's Ricci flow, it is important to study self-similar solutions, i.e., solutions that change only by diffeomorphisms and rescaling. The simplest self-similar solutions are the gradient solitons which associate canonical solutions to the RH flow, see [11]. There are two methods to study gradient solitons: one is to treat them as static manifolds, and the other is to associate them with canonical solutions to the flow.…”

“…In addition to sharing many good properties with the Ricci flow, the RH flow is less singular than the Ricci flow or the harmonic map flow alone. See [11] for details, as well as [8,13] for results and examples of this coupled flow. In [6], we developed a stochastic approach to the harmonic map heat flow on manifolds with time-dependent metric; there, however, the evolution of the metric is not coupled to the heat flow.…”

“…The following definition is taken from [11]. Recently, there have been some papers generalizing gradient Ricci solitons, for instance [2,12] on almost Ricci solitons, [3] on quasi-Einstein manifolds and [7] on generalized quasi-Einstein manifolds.…”

“…In [11], Müller introduced a new flow generalizing Hamilton's Ricci flow, called the Ricci flow coupled with harmonic map heat flow, which we call Ricci-Harmonic flow or RH flow for short in the sequel. In addition to sharing many good properties with the Ricci flow, the RH flow is less singular than the Ricci flow or the harmonic map flow alone.…”