  Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature
flow) entropy for submanifolds in a general ambient Riemannian manifold. In
particular, this entropy is equivalent to area growth of a closed submanifold
in a closed ambient manifold with non-negative Ricci curvature. Moreover, this
entropy is monotone along the mean curvature flow in a closed Riemannian
manifold with non-negative sectional curvatures and parallel Ricci curvature.
As an application, we show the partial regularity of the limit of mean
curvature flow of surfaces in a three dimensional Riemannian manifold with
non-negative sectional curvatures and parallel Ricci curvature.
