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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...

physixfan
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    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? – ACuriousMind Jan 14 '15 at 23:01
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    But can we write a curved surface as f(x,y) in 2D situation? – physixfan Jan 14 '15 at 23:02
  • Like f=x^2-y^2 is a curved surface but with 0 mean curvature... – physixfan Jan 14 '15 at 23:03
  • Like in 1D, the second order derivative is the curvature for a line... – physixfan Jan 14 '15 at 23:04
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    Yes, you can write the height of a 2D membrane as f(x,y) given the curvature is not too large. Similarly, if the curvature is not too large, you can use the Monge gauge and approximate the curvature using the Laplacian. – alarge Jan 14 '15 at 23:06
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    Essentially a duplicate of http://physics.stackexchange.com/q/20714/2451 and links therein. – Qmechanic Jan 14 '15 at 23:06
  • @Qmechanic it's not a duplicate. I know these two are related and similar, but I am wondering whether they are exactly the same.. if not, what are the differences... – physixfan Jan 14 '15 at 23:10
  • @alarge Thanks.. I am trying to learn what is a Monge gauge... – physixfan Jan 14 '15 at 23:11
  • See for example http://www.cmu.edu/biolphys/deserno/pdf/membrane_theory.pdf, eq. (20) and the "Box 5" that follows it. – alarge Jan 14 '15 at 23:13
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    @alarge Thanks for your information! I understand now... – physixfan Jan 14 '15 at 23:26

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