Ancient mean curvature flows with finite total curvature

We construct an I-family of ancient graphical mean curvature flows over a minimal hypersurface in Rⁿ⁺¹ of finite total curvature with the Morse index I by establishing exponentially fast convergence in terms of |x|²-t. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.