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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...

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    $\begingroup$ What? Curvature is a property of a manifold, and the Laplace operator is an operator on smooth functions - how could these possibly be the same? $\endgroup$
    – ACuriousMind
    Jan 14 '15 at 23:01
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    $\begingroup$ But can we write a curved surface as f(x,y) in 2D situation? $\endgroup$
    – physixfan
    Jan 14 '15 at 23:02
  • $\begingroup$ Like f=x^2-y^2 is a curved surface but with 0 mean curvature... $\endgroup$
    – physixfan
    Jan 14 '15 at 23:03
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    $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/20714/2451 and links therein. $\endgroup$
    – Qmechanic
    Jan 14 '15 at 23:06
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    $\begingroup$ @alarge Thanks for your information! I understand now... $\endgroup$
    – physixfan
    Jan 14 '15 at 23:26

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