Mean Curvature Flow in an Extended Ricci Flow Background

We consider a functional related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying mean curvature flow in a Ricci flow background. This functional is the 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. In this time-derivative formula, an extension of Hamilton's differential Harnack expression on the boundary integrand appears. 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 Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Finally, we present examples of mean curvature solitons in an extended Ricci flow background.