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

Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces