From this question, I know the mean curvature flow is not strictly parabolic. For proving the short time existence ,by trick of De Turck, there is $$ \partial_t \bar X =g^{ij}(\partial_i\partial_j \bar X -\bar\Gamma_{ij}^k \partial_k \bar X) $$ But why it is strictly parabolic ? I think I can do same thing as in question.
Besides, why $\Delta_{g(t)} \bar X = \Delta_{g(t)} X$ in the middle part of picture below?
