I'm currently reading the paper https://arxiv.org/pdf/1007.3899.pdf and I have a question regarding the mean curverture stuff in it (unfortunately, I don't have any knowledge in differential geometry)- Lemma 3.7.
The situation is the following:
Let $E\subset \mathbb{R}^n$ be a Borel set with positive and finite Lebesgue measure. Even better we consider $\partial E$ to be of class $C^{1,\eta}(B_1(0))$ for all $\eta \in (0,1)$ and such that $\partial E$ is a nonparametric surface in $\mathbb{R}^{n-1}$, i.e. a graph of a function $f\in C^1(B_1(0))$ (see theorem 3.2 (ii) ).
How is the scalar mean curvature of $\partial E$ with orientation induced by the inner normal to E defined? (see lemma 3.7)
I already 'found out' that the scalar mean curvature of $\partial E$, $H$, is defined as $$H:=\frac{1}{n-1}\text{div}\big(\frac{\nabla f(x)}{\sqrt{1+\|\nabla f(x)\|^2 }} \big).$$
And what is the inner normal to $E$, is that $\nu =-\frac{\nabla f(x)}{\sqrt{1+\|\nabla f(x)\|^2 }}$?
I guess that the scalar mean curvature of $\partial E$ with orientation induced by the inner normal to E is $-H$?
Thank you.