Let $M$ be a constant mean curvature surface / $n$-manifold in $\mathbb{R}^{n+1}$, with mean curvature $H$ and second fundamental form $A$. Let $N$ be the Gauss map (unit normal) for $M$ and $X$ be the position / coordinate vector. Let $\triangle_M$ denote the Laplace-Beltrami operator on $M$.

It is well known that $$\triangle_M X = (nH) N.$$ It is sorta well known that $$\triangle_M N = -|A|^2 N.$$ (Note that $|A|^2$ is also the sum of the squares of the principle curvatures if that helps.)

But I remember a third for which I cannot find a reference: $$\triangle (N \cdot X) = - |A|^2 (N \cdot X)$$

I think this correct but if someone could give me a hint or link I would be most appreciative.

  • $\begingroup$ Could you please provide a reference for the first result? $\endgroup$ – hyperkahler Jun 4 '17 at 0:29

It seems that there is one more term:

$$\begin{split} \Delta (N\cdot X) &= (\Delta N, X) + (N, \Delta X) + 2 (\nabla N, \nabla X) \\ &= (-|A|^2 N, X) + (N, (nH) N) + 2 \sum_{i,j} (-A_{ij}e_j, e_i)\\ &= -|A|^2(N, X) + nH - 2\sum_i A_{ii}\\ &= -|A|^2 (N, X) - nH. \end{split}$$


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