Regarding Lemma 9.3 (page 262) in this paper by Huisken, I have two questions.

  1. It is said $$\vert \tilde{M}_{\tilde t} \vert = \frac{1}{n}\int \tilde H (\tilde F \tilde \nu) d\mu$$ follows from the 1st variation formula.

I can only recognize the integral is the variation of area functional, but I don't know how to derive this equation from the 1st variation formula.

  1. Why the enclosed volume $\tilde V$ can be estimated by ball of radius $(\varepsilon \tilde H_{\text{min}})^{-1}$, knowing its second fundamental form $\tilde h_{ij} \geq \varepsilon \tilde H_{\text{max}}\tilde g_{ij}$ ? This is reasonable on an intuitive level, but how can I write down an explicit proof ?



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