Mean curvature flow in an extended Ricci flow background

In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of them is a weighted extended version of the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman's modified extended Ricci flow. For instance, the boundary integrand term provides an extension of Hamilton's differential Harnack expression for mean curvature flows in Euclidean space. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish a Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Also, we show how for constructing examples of mean curvature solitons in an extended Ricci flow background.