From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I encounter some problem in the derivation of Lemma $5.2$. Here is my steps,

$$\frac{\partial}{\partial t} \left( \frac{|A|^2}{H^{\alpha}} - \frac{1}{n} H^{2 - \alpha} \right)$$

By using chain rule and quotient rule,

$$= \frac{H^{\alpha}\partial_t |A|^2-\alpha |A|^2H^{\alpha-1} \partial _t H }{H^{\alpha+2}}- \frac{1}{n} (2 - \alpha)H^{1-\alpha}\partial_t H$$

Since we know that $\partial_t |A|^2 = \Delta |A|^2 - 2 |\nabla A|^2 + 2 |A|^4$ and we also know that $\partial_t H = \Delta H + |A|^2 H,$ I try to plug in and I got the term with $H$ to the power of $-3$ and $-2.$ But in the paper, the term has power $-\alpha$ or $-\alpha-1.$ My guess might be that the way I use chain rule and quotient rule was wrong.

Can someone spot my issue?


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