Let's assume we study 2D function/surface f(x,y).
Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$
And the mean curvature: let $\kappa_1$ and $\kappa_2$ be the principal curvatures, then the mean curvature is defined as: $$H=\frac12(\kappa_1+\kappa_2)$$
We can ignore the coefficient $\frac12$ for discussion. The question is: are these two exactly the same for 2D functions? If not what is the difference? I am not quite familiar with differential geometry...
