Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

Abstract

The main objective of this thesis is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with the Ricci flow coupled with the harmonic map heat flow and calculate its variation along parameters that preserve the weighted volume measure. So, an extension of the Harnack-Hamilton differential appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. As an application, we consider the associated normalized family of the mean curvature flow to obtain results of convergence in the Cheeger-Gromov sense in the compact and noncompact cases. Moreover, we show how to construct a family of mean curvature solitons and we establish a characterization of such a family