Mean curvature after reflection

Question

Why does the mean curvature at a point on a surface stay the same if we reflect the surface across a plane? Thank you.

Answer 1

Intuitively, if you look at a surface in a mirror, the surface normal and principal sections are reflected "compatibly", so the sign of the mean curvature is unchanged.

In more detail, it suffices to show that if $S$ is a regular surface in Euclidean three-space, $p$ is a point of $S$, and $R$ is a reflection, then the mean curvature of $R(S)$ at $R(p)$ is equal to the mean curvature of $S$ at $p$. Since translations and rotations obviously preserve mean curvature, we may as well assume $R$ is reflection in a plane through $p$ containing the normal to $S$ at $p$ and one of the principal directions. This reflection preserves the quadratic approximation of $S$ at $p$, so it preserves the mean curvature of $S$ at $p$. Since $p$ was arbitrary, reflections preserve the mean curvature function of $S$.