I am not an expert in differential geometry and was wondering: Given a function $h:\mathbb S^{n-1} \longrightarrow \mathbb R $ smooth, the graph of $h$ is an $(n-1)$-dimensional submanifold of $\mathbb R^n$. How can I compute the mean curvature of this submanifold in terms of $h$ with regards to the metric on the sphere?
Here, $\mathbb S^{n-1} := \{z \in \mathbb R^n: |z| = 1\}$ is the unit sphere seen as a smooth Riemannian submanifold of $\mathbb R^n$.
What I am especially interested in is the case $n=3$ and I am looking for an explicit formula.
Can anyone provide an answer/formula, or a reference?
Thank you guys very much for your help.